UC-NRLF 


$B    527    fibD 


-rSXHif,  >  ]iJ-V^-KxawiMil&»it 


*(tW«*.'.''.-J^KV>'4.'VWS*Tf«V«. 


IGONOMETI\Y 


Lyman  and  Goddars 


WOilW;j«<rwiHjiWMWH»limi|IW»li'IMIllli|ll 


tkif.i-  -    -■>-:a^-gy,'vity;^T»w^'W«|iiiwina'Hill>m*rl>i 


■a««»»»wvTrocwi>«iduoaM>w«>>w»Mw*»viwnoqww«twiWrt»WWi<)UCts^ 


With  Tables 


PLANE  TRIGONOMETRY 


BY 

ELMER   A.    LYMAN 

MICHIGAN  STATE   NORMAL  COLLEGE 
AND 

EDWIN   C.    GODDARD 

UNIVERSITY  OF  MICHIGAN 


3»<C 


ALLYN    AND    BACON 

Boston  anti  Chicago 


Copyright,  1899,  by 
ELMER  A.  LYMAN  and  EDWIN  C.  GODDARD. 


Korisooti  ^rees 

J.  S.  Gushing  &  Co.  —  Berwick  &  Smith 
Norwood  Mass.  U.S.A. 


PREFACE. 

The  need  felt  by  tlie  authors  in  their  class-room  for  a  text-book 
furnishing  sufficient  material  in  analytical  trigonometry,  and  also 
in  the  solution  of  the  triangle,  is  responsible  for  the  appearance 
of  this  book.  American  text-books,  for  the  most  part,  treat  this 
latter,  practical,  part  of  the  subject  fully;  English  text-books 
elaborate  the  former,  theoretical,  part ;  but  no  book  available 
seems  to  meet  both  needs  adequately.  To  do  that  is  the  first  aim 
of  the  present  work.  Nearly  everything  in  the  book  has  been 
worked  out  in  the  class-room,  and  tried  by  that  sure  test. 

Once  under  way  the  work  grew,  and  other  features  demanded 
attention.  For  some  unaccountable  reason  nearly  all  books,  in 
the  proof  of  the  formulae  for  functions  of  a  ±  p,  treat  the  same 
line  as  both  positive  and  negative  in  the  same  discussion,  thus 
vitiating  the  proof ;  and  in  many  cases  proofs  are  given  for  acute 
angles,  and  are  then  supposed  to  be  established  without  further 
discussion  for  all  angles.  Some  books,  indeed,  suggest  that  the 
student  can  draw  other  figures  and  show  that  the  formula  holds 
in  all  cases.  As  a  matter  of  fact  the  student  cannot  show  any- 
thing of  the  kind ;  and  if  he  could,  the  proof  would  still  apply 
only  to  conditions  the  same  as  in  those  figures  actually  drawn, 
and  not  to  all  the  other  indefinite  number  of  possible  combina- 
tions of  conditions.  These  difficulties  have  been  avoided  by  so 
stating  the  proofs  that  the  language  applies  to  figures  involving 
any  angles,  and  to  avoid  drawing  an  indefinite  number  of  such 
figures,  as  would  be  necessary  fully  to  establish  the  formulae 
geometrically,  resort  has  been  made  to  the  algebraic  proof  for  the 
general  case  (see  page  58). 

Inverse  functions  have  been  introduced  early,  and  used  through- 
out the  work,  so  as  to  familiarize  the  student  with  that  important 

800555 


iv  PREFACE. 

notation.  From  the  beginning,  wherever  computations  are  intro- 
duced they  are  made  by  means  of  logarithms.  The  average  stu- 
dent, using  logarithms  for  a  short  time  and  only  at  the  end  of 
the  subject,  goes  away  and  straightway  forgets  what  manner  of 
things  they  are.  It  is  hoped,  by  dint  of  much  practice,  extended 
over  as  long  a  time  as  possible,  to  give  the  student  a  command 
of  logarithms  that  will  stay.  The  fundamental  formulae  of  trigo- 
nometry must  be  memorized.  There  is  no  substitute  for  this. 
To  assist  in  thus  fixing  formulae  in  mind,  considerable  oral  work 
has  been  introduced,  and  frequent  lists  of  review  problems  in- 
volving all  principles  and  formulae  previously  developed.  These 
lists  serve  the  further  purpose  of  throwing  the  student  on  his 
own  resources,  and  compelling  him  to  find  in  the  problem  itself, 
and  not  in  any  model  solution,  the  key  to  its  solution,  thus  devel- 
oping power,  instead  of  mere  ability  to  imitate.  Enough  prob- 
lems are  provided  so  that  different  selections  may  be  assigned  to 
different  members  of  a  class,  or  to  classes  in  different  years.  It 
is  not  expected  that  each  student  will  be  able  to  solve  all  the 
problems  in  the  time  usually  given  to  the  subject.  Articles 
marked  *  {see  Art.  *26)  may  be  omitted  unless  the  teacher  finds 
time  for  them  without  7ieglecting  the  rest  of  the  work.  Do  not  assign 
too  much  work  at  first.  Make  sure  the  student  has  complete  mastery 
of  the  fundamental  formulce. 

Special  attention  is  called  to  the  fact  that  in  the  solution  of 
triangles,  divisions  and  subdivisions  into  cases  have  been  aban- 
doned, and  the  student  is  thrown  on  his  own  resources  to  select 
from  the  three  possible  sets  of  formulae  those  leading  to  the  solu- 
tions from  the  given  data.  Long  experience  has  shown  that  this 
tends  to  clearness  and  simplicity.  The  use  of  checks  is  insisted 
upon  in  all  computations. 

No  complete  acknowledgment  of  help  received  could  here  be 
made.  The  authors  are  under  obligation  to  many  who  have  con- 
tributed general  hints,  and  to  several  who,  after  going  over  the 
manuscript  and  proof  with  care,  have  given  valuable  suggestions. 
The  standard  works  of  Levett  and  Davison,  Hobson,  Henrici  and 
Treutlein,  and  others,  have  been  freely  consulted,  and  while  many 
of  the  problems  have  been  prepared  by  the  authors  in  their  class- 


PREFACE.  V  ( 

room  work,  they  have  not  hesitated  to  take,  from  such  standard 

collections  as  writers  generally  have  drawn  upon,  any  problems  \ 

that  seemed  better  adapted  than  others  to  the  work.     Quality  ] 

has  not  been  knowingly  sacrificed  to  originality  in  making  this  j 

book.     Corrections  and  suggestions  will  be  gladly  received  at  any  i 

time.  1 

E.  A.  L.  i 

E.  C.  G.  5 

OCTOBEB,  1899.  1 


CONTENTS. 

Chapter  I.  Angles — Measurement  of  Angles. 


^.^^ 


PAGE 

Angles ;  magnitude  of  angles 1 

Rectangular  axes ;  direction       .        .        .      * 2 

Measurement;  sexagesimal  and  circular  systems  of  measurement; 

the  radian 3 

Examples 6 

Chapter  II.     The  Trigonometric  Functions. 

Function  defined 8 

The  trigonometric  functions 9 

Fundamental  relations 11 

Examples 14 

Functions  of  0°,  30°,  45°,  60°,  90° 15 

Examples 18 

Variations  in  the  trigonometric  functions 19 

Graphic  representation  of  functions ,        .22 

Examples 27 

Chapter  in.  Functions  of  any  Angle — Inverse 
Functions. 

Relations  of  functions  of   -  0,  90°  ±  B,  180°  ±  Q,  270°  ±  ^  to  the 

functions  of  ^ 29 

Inverse  functions ,        ,        ,        ,        .35 

Examples 36 

Review 38 

Chapter  IV.    Computation  Tables. 

Natural  functions 40 

Logarithms 40 

Laws  of  logarithms 42 

Use  of  tables 45 

Cologarithms 49 

Examples 50 

vii 


viii  CONTENTS. 

Chapter  V.    Applications. 


PA6B 


Measurements  of  heights  and  distances 51 

Common  problems  in  measurement 52 

Examples 54 

Chapter  VI.    General  Formula.  —  Trigonometric  Equa- 
tions AND  Identities. 

Sine,  cosine,  tangent  of  a  ±  )8     .        . 56 

Examples    .         .         .         ',        .        .        ....        .         .59 

Sin  0  ±  sin  cfy,  cos  6  ±  cos  ^ 61 

Examples 62 

Functions  of  the  double  angle    .        .        .        .        ...        .63 

Functions  of  the  half  angle .        .64 

Examples ,        .         .64 

Trigonometric  equations  and  identities 66 

Method  of  attack 66 

Examples     ... 67 

Simultaneous  trigonometric  equations 69 

Examples .        .        .        .        .70 

Chapter  VII.    Triangles. 

Laws  of  sines,  tangents,  and  cosines  . 72 

Area  of  the  triangle 76 

Solution  of  triangles 76 

Ambiguous  case 78 

Model  solutions 80 

Examples 83 

Applications 84 

Review 86 


PLANE   TRIGONOMETRY. 

CHAPTER  I. 

ANGLES  — MEASUREMENT   OF   ANGLES. 

1.  Angles.  It  is  difficult,  if  not  impossible,  to  define  an 
angle.  This  difficulty  may  be  avoided  by  telling  how  it 
is  formed.  If  a  line  revolve  about  one  of  its  points^  an  angle 
is  generated^  the  magnitude  of  the  angle  depending  on  the 
amount  of  the  rotation. 

Thus,  if  one  side  of  the  angle  ^,  as  OR^  be  originally  in 
the  position  OX^  and  be  revolved  about  the  point  0  to  the 
position  in  the  figure,  the 
angle  XOR  is  generated. 
OX  is  called  the  initial  line, 
and  any  position  of  OR  the 
terminal  line  of  the  angle 
formed.  The  angle  6  is 
considered  positive  if  gener- 
ated hy  a  counter-clockwise 
rotation  of  OR,  and  hence  negative  if  generated  hy  a  clockwise 
rotation.  The  magnitude  of  0  depends  on  the  amount  of 
rotation  of  OR,  and  since  the  amount  of  such  rotation  may 
be  unlimited,  there  is  no  limit  to  the  possible  magnitude  of 
angles,  for,  evidently,  the  revolving  line  may  reach  the  posi- 
tion OR  by  rotation  through  an  acute  angle  6,  and,  likewise, 
by  rotation  through  once,  twice,  •••,  w  times  360°,  plus  the 
acute  angle  6,  So  that  XOR  may  mean  the  acute  angle 
(9,  ^  -f-  360°,  e  +  720°,  .-,  O-^n-  360°. 

1 


i?>' 


Fig.  1. 


£  PLANE   TRIGONOMETRY. 

In  reading  an  angle,  read  first  the  initial  line,  then  the 
terminal  line.  Thus  in  the  figure  the  acute  angle  XOE,  or 
xr,  is  a  positive  angle,  and  ROX^  or  rx^  an  equal  negative 
angle. 

Ex.  1.  Show  that  if  the  initial  lines  for  \,  f,  ^/,  —  ^,  right  angles  are 
the  same,  the  terminal  lines  may  coincide. 

2.  Name  four  other  angles  having  the  same  initial  and  terminal  lines 
as  ^  of  a  right  angle ;  as  f  of  a  right  angle ;  as  f  of  a  right  angle. 

2.  Rectangular  axes.  Any  plane  surface  may  be  divided 
by  two  perpendicular  straight  lines  XX^  and  YY'  into  four 

portions,  or  quadrants. 

XX'  is  known  as  the  x-axis, 
YY'  as  the  y-axis^  and  the  two 
together  are  called  axes  of  refer- 

X     ence.     Their  intersection  0  is  the 

origin^  and   the   four   portions   of 
the  plane  surface,  XOY,    YOX', 
y"  X'OY',  Y'OX,  are  called  respec- 

Fjq,  2.  tively  the  first,  second,  third,  and 

fourth  quadrants.  The  position  of 
any  point  in  the  plane  is  determined  when  we  know  its  dis- 
tances and  directions  from  the  axes. 

3.  Any  direction  may  be  considered  positive.     Then  the 
opposite  direction  must  be  negative.    Thus,  if  AB  represents 
any  positive  line,  BA  is  an  equal  nega- 
tive  line.       Mathematicians    usually 

consider  lines  measured  in  the  same  direction  as  OX  or  OY 
(Fig.  2)  as  positive.  Then  lines  measured  in  the  same  direc- 
tion as  OX'  or  OY'  must  he  negative. 

The  distance  of  any  point  from  the  ?/-axis  is  called  the 
abscissa,  its  distance  from  the  a;-axis  the  ordinate,  of  that 
point ;  the  two  together  are  the  coordinates  of  the  point, 
usually  denoted  by  the  letters  x  and  y  respectively,  and 
written  (x,  y). 


Y 

p' 

P 

,/ 

0 

N 

p" 

Y' 

P'" 

ANGLES  —  MEASUREMENT.  3 

When  taken  with  their  proper  signs,  the  coordinates  define  completely 
the  position  of  the  point.  Thus,  if  the  point  P  is  +  a  units  from  YY', 
and  +  h  units  from  XX',  any  convenient 
unit  of  length  being  chosen,  the  position  of 
P  is  known.  For  we  have  only  to  measure 
a  distance  ON  equal  to  a  units  along  OX, 
and  then  from  N  measure  a  distance  h 
units  parallel  to  OY,  and  we  arrive  at  the 
position  of  the  point  P,  (a,  &).  In  like 
manner  we  may  locate  P',  (—  «,  ^),  in  the 
second  quadrant,  P",  (—a,  —  Z>),  in  the 
third    quadrant,    and    P'",    (a,    —  &),    in  „      ^ 

the  fourth  quadrant. 

Ex.  Locate  (2,  -2);  (0,0);  (-8,  -7);  (0,  5);  (-2,  0);  (2,  2); 
(m,  n). 

4.  If  OX  is  the  initial  line,  0  is  said  to  be  an  angle  of  the 
first,  second,  third,  or  fourth  quadrant,  according  as  its  ter- 
minal line  is  in  the  first,  second,  third,  or  fourth  quadrant. 
It  is  clear  that  as  OR  rotates  its  quality  is  in  no  way  affected, 
and  hence  it  is  in  all  positions  considered  positive,  and  its  ex- 
tension through  0,  OB',  negative. 

The  student  should  notice  that  the  initial  line  may  take  any  position 
and  revolve  in  either  direction.  While  it  is  customary  to  consider  the 
counter-clockwise  rotation  as  forming  a  positive  angle,  yet  the  condi- 

V.  ^'^'    tions  of  a  figure  may  be  such 
•     /    that  a  positive   angle  may  be 
\/      generated  by  a  clockwise  rota- 
yu       ^     ^        ""^B'  /V    /4^       *io^-     Thus  the  angle  X072  in 
-2^'    j^'r    \x     each  figure  may  be  traced  as 
p       .  a  positive   angle  by  revolving 

the  initial  line  OX  to  the  posi- 
tion OR.  No  confusion  can  result  if  the  fact  is  clear  that  when  an 
angle  is  read  XOP,  OX  is  considered  a  positive  line  revolving  to  the 
position  OR.  OX'  and  OR'  then  are  negative  lines  in  whatever  direc- 
tions drawn.  These  conceptions  are  mere  matters  of  agreement,  and  the 
agreement  may  be  determined  in  a  particular  case  by  the  conditions  of 
the  problem  quite  as  well  as  by  such  general  agreements  of  mathema- 
ticians as  those  referred  to  in  Arts.  3  and  4  above. 

5.  Measurement.  All  measurements  are  made  in  terms 
of  some  fixed  standard  adopted  as  a  unit.     This  unit  must 


4  PLANE   TRIGONOMETRY. 

be  of  the  same  kind  as  the  quantity  measured.  Thus,  length 
is  measured  in  terms  of  a  unit  length,  surface  in  terms  of  a 
unit  surface,  weight  in  terms  of  a  unit  weight,  value  in  terms 
of  a  unit  value,  an  angle  in  terms  of  a  unit  angle. 

The  measure  of  a  given  quantity  is  the  number  of  times  it 
contains  the  unit  selected. 

Thus  the  area  of  a  given  surface  in  square  feet  is  the 
number  of  times  it  contains  the  unit  surface  1  sq.  ft. ;  the 
length  of  a  road  in  miles,  the  number  of  times  it  contains 
the  unit  length  1  mi. ;  the  weight  of  a  cargo  of  iron  ore  in 
tons,  the  number  of  times  it  contains  the  unit  weight  1  ton ; 
the  value  of  an  estate,  the  number  of  times  it  contains  the 
unit  value  f  1. 

The  same  quantity  may  have  different  measures,  according 
to  the  unit  chosen.  So  the  measure  of  80  acres,  when  the 
unit  surface  is  1  acre,  is  80,  when  the  unit  surface  is  1  sq.  rd., 
is  12,800,  when  the  unit  surface  is  1  sq.  yd.,  is  387,200. 
What  is  its  measure  in  square  feet  ? 

6.  The  essentials  of  a  good  unit  of  measure  are : 

1.  That  it  be  invariable,  i.e.  under  all  conditions  bearing 
the  same  ratio  to  equal  magnitudes. 

2.  That  it  be  convenient  for  practical  or  theoretical  pur- 
poses. 

3.  That  it  be  of  the  same  kind  as  the  quantity  measured. 

7.  Two  systems  of  measuring  angles  are  in  use,  the  sexa- 
gesimal and  the  circular. 

The  sexagesimal  system  is  used  in  most  practical  applica- 
tions. The  right  angle,  the  unit  of  measure  in  geometry, 
though  it  is  invariable,  as  a  measure  is  too  large  for  con- 
venience. Accordingly  it  is  divided  into  90  equal  parts, 
called  degrees.  The  degree  is  divided  into  60  minutes,  and 
the  minute  into  60  seconds.  Degrees,  minutes,  seconds,  are 
indicated  by  the  marks  °  '  ",  as  36°  20'  15''. 

The  division  of  a  right  angle  into  hundredths,  with  subdivisions  into 
hundredths,  would  be  more  convenient.    The  French  have  proposed  such 


MEASUREMENT   OF   ANGLES. 


a  centesimal  system,  dividing  the  right  angle  into  100  grades,  the  grade 
into  100  minutes,  and  the  minute  into  100  seconds,  marked  ^^  ''\  as  508 
70^  28^\  The  great  labor  involved  in  changing  mathematical  tables, 
instruments,  and  records  of  observation  to  the  new  system  has  prevented 
its  adoption. 

8.    The  circular  system  is  important  in  theoretical  con- 
siderations.    It  is  based  on  the  fact  that  for  a  given  angle 
the  ratio  of  the  length  of  its  arc  to  the  length  of  the  radius 
of  that  arc  is  constant,  i.e.  for  a  fixed 
angle  the  ratio  arc :  radius  is  the  same 
no    matter   what    the   length    of   the 
radius.     In  the  figure,  for  the  angle  ^, 

OA       OB       OQ 


AA'     BB'      CQ' 

That  this  ratio  of  arc  to  radius  for  a  fixed  angle  is  constant 
follows  from  the  established  geometrical  principles : 

1.  The  circumference  of  any  circle  is  2  tt  times  its  radius. 

2.  Angles  at  the  centre  are  in  the  same  ratio  as  their  arcs. 

The  Radian.  It  follows  that  an  angle  whose  arc  is  equal 
in  length  to  the  radius  is  a  constant  angle  for  all  circles, 
since  in  four  right  angles,  or  the  perigon,  there  are  always 
2  7r  such  angles.  This  constant  angle., 
ivhose  arc  is  equal  in  length  to  the  radius., 
is  taken  as  the  unit  angle  of  circular 
measure.,  and  is  called  the  radian.  From 
the  definition  we  have 


4  right  angles  =  360° 
2  right  angles  =  180° 


2  TT  radians, 
TT  radians. 


Fig.  6. 


TT 


1  right  angle   =    90°  =  —  radians. 


TT  is  a  numerical  quantity,  3.14159+,  and  not  an  angle.      When  we 
speak  of  180°  as  tt,  90°  as  ^,  etc.,  we  always  mean  tt  radians,  ^  radians,  etc. 


6  PLANE  TRIGONOMETRY. 

9.  To  change  from  one  system  of  measurement  to  the 
other  we  use  the  relation, 

2  TT  radians  =  360°. 

.  •.  1  radian  =  i^  =  57^.2958-  ; 

TT 

i.e.  the  radian  is  57°.3,  approximately. 

Ex.  1.   Express  in  radians  75°  30'. 

75°  30'  =  75°.5 ;  1  radian  =  57°.3. 

.-.  75°  30'  =  —  =  1.317  radians. 
57.3 

2.   Express  in  degree  measure  3.6  radians. 
1  radian  =  57°.3. 
.-.  3.6  radians  =  3.6  x  o7°.3  =  206°  16'  48". 

EXAMPLES. 

1.  Construct,  approximately,  the  following  angles :  50°,  —  20°,  90°, 
179°,    -135°,  400°,   -380^  1140°,   |  radians,    |  radians,    --radians, 

q  -If) 

3  IT  radians,    — ^  radians,   — — ^  radians.      Of  which  quadrant  is  each 
angle?  ^  ^ 

2.  What  is  the  measure  of : 

(a)  f  of  a  right  angle,  when  30°  is  the  unit  of  measure  ? 

(b)  an  acre,  when  a  square  whose  side  is  10  rds.  is  the  unit  ? 

(c)  m  miles,  when  y  yards  is  the  unit  ? 

3.  What  is  the  unit  of  measure,  when  the  measure  of  2^  miles  is  50? 

4.  The  Michigan  Central  R.R.  is  535  miles  long,  and  the  Ann  Arbor 
R.R.  is  292  miles  long.  Express  the  length  of  the  first  in  terms  of  the 
second  as  a  unit. 

5.  What  will  be  the  measure  of  the  radian  when  the  right  angle  is 
taken  for  the  unit  ?     Of  the  right  angle  when  the  radian  is  the  unit  ? 

6.  In  which  quadrant  is  45°?  10°?  -60°?  145°?  1145°?  -725°? 
Express  each  in  right  angles ;  in  radians. 

7.  Express  in  sexagesimal  measure 

J,  ^.  1,  6.28,  I,  1^,  -i^,  radians. 

O      12  TT       o  3 


EXAMPLES.  7 

8.  Express  in  each  system  an  interior  angle  of  a  regular  hexagon ; 
an  exterior  angle. 

9.  Find  the  distance  in  miles  between  two  places  on  the  earth's 
equator  which  are  11°  15'  apart.    (The  earth's  radius  is  about  3963  miles.) 

10.  Find  the  length  of  an  arc  which  subtends  an  angle  of  4  radians 
at  the  centre  of  a  circle  of  radius  12  ft.  3  in. 

11.  An  arc  15  yds.  long  contains  3  radians.  Find  the  radius  of  the 
circle. 

12.  Show  that  the  hour  and  minute  hands  of  a  watch  turn  through 
angles  of  30^  and  6°  respectively  per  minute ;  also  find  in  degrees  and  in 
radians  the  angle  turned  through  by  the  minute  hand  in  3  hrs.  20  mins. 

13.  Find  the  number  of  seconds  in  an  arc  of  1  mile  on  the  equator ; 
also  the  length  in  miles  of  an  arc  of  1'  (1  knot). 

14.  Find  to  three  decimal  places  the  radius  of  a  circle  in  which  the 
arc  of  71°  36'  3''.6  is  15  in.  long. 

15.  Find  the  ratio  of  -  to  5°. 

6 

16.  What  is  the  shortest  distance  measured  on  the  earth's  surface 
from  the  equator  to  Ann  Arbor,  latitude  +  42°  16'  48"? 

17.  The  difference  of  two  angles  is  10°,  and  the  circular  measure  of 
their  sum  is  2.     Find  the  circular  measure  of  each  angle. 

18.  A  water  wheel  of  radius  6  ft.  makes  30  revolutions  per  minute. 
Find  the  number  of  miles  per  hour  travelled  by  a  point  on  the  rim. 


CHAPTER   II. 

THE  TRIGONOMETRIC  FUNCTIONS. 

10.  Trigonometry,  as  the  word  indicates,  was  originally 
concerned  with  the  measurement  of  triangles.  It  now 
includes  the  analytical  treatment  of  certain  functions  of 
angles,  as  well  as  the  solution  of  triangles  by  means  of  cer- 
tain relations  between  the  functions  of  the  angles  of  those 
triangles. 

11.  Function.  If  one  quantity  depends  upon  another  for 
its  value,  the  first  is  called  a  function  of  the  second.  It 
always  follows  that  the  second  quantity  is  also  a  function  of 
the  first ;  and,  in  general,  functions  are  so  related  that  if  one 
is  constant  the  other  is  constant,  and  if  either  varies  in  value, 
the  other  varies.  This  relation  may  be  extended  to  any 
number  of  mutually  dependent  quantities. 

Illustration.  If  a  train  moves  at  a  rate  of  30  miles  per 
hour,  the  distance  travelled  is  a  function  of  the  rate  and 
time,  the  time  is  a  function  of  the  rate  and  distance,  and  the 
rate  is  a  function  of  the  time  and  distance. 

Again,  the  circumference  of  a  circle  is  a  function  of  the 
radius,  and  the  radius  of  the  circumference,  for  so  long  as 
either  is  constant  the  other  is  constant,  and  if  either  changes 
in  value,  the  other  changes,  since  circumference  and  radius 
are  connected  by  the  relation  (7=2  irR. 

Once  more,  in  the  right  triangle 

NOP,  the  ratio  of  any  two  sides  is 

a  function  of  the  angle  a,  because 

p"  N'  N      ^      ^^^  ^^®  right  triangles  of  which  a  is 

FiQ.  7.  one  angle  are  similar,  i,e.  the  ratio 

8 


THE   TRIGONOMETRIC   FUNCTIONS. 


9 


of  two  corresponding  sides  is  constant  so  long  as  a.  is  con- 
stant, and  varies  if  «  varies. 
Thus,  the  ratios 

NP  ^  N'P'  ^  WP'' 
OP 


and 


ojsr 

NP 


OP' 

ON' 

N'P' 


OP" 

ON"       , 


depend  on  a  for  their  values,  i.e.  are  functions  of  a. 

12.  The  trigonometric  functions.  In  trigonometry  six 
functions  of  angles  are  usually  employed,  called  the  trigono- 
metric functions. 

By  definition  these  functions  are  the  six  ratios  between  the 
sides  of  the  triangle  of  reference  of  the  given  angle.  The 
triangle  of  reference  is  formed  by  drawing,  from  some  point  in 
the  initial  line.,  or  the  initial  line  produced^  a  perpendicular  to 
that  line  meeting  the  terminal  line  of  the  angle. 


Fig.  8. 


Let  a  be  an  angle  of  any  quadrant.  Each  triangle  of 
reference  of  a,  NOP,  is  formed  by  drawing  a  perpendicular 
to  OX,  or  OX  produced,  meeting  the  terminal  line  OB  in  P. 


10  PLANE   TRIGONOMETRY. 

If  «  is  greater  than  360°,  its  triangle  of  reference  would 
not  differ  from  one  of  the  above  triangles. 

It  is  perhaps  worthy  of  notice  that  the  triangle  of  reference  might  be 
defined  to  be  the  triangle  formed  by  drawing  a  perpendicular  to  either 
side  of  the  angle,  or  that  side  produced,  meet- 
ing the  other  side  or  the  other  side  produced. 
In  the  figure,  NOP  is  in  all  cases  the  triangle 
of  reference  of  a.     The  principles  of  the  fol- 


N 


\       "^    ,''0         P     2f 

j    ,,.-'jv-  lowing  pages  are  the  same  no  matter  which 

^''P  of  the  triangles  is  considered  the  triangle  of 

Fig.  9.  reference.     It  will,  however,  be  as  well,  and 

perhaps  clearer,  to  use  the  triangle   defined 

under  Fig.  8,  and  we  shall  always  draw  the  triangle  as  there  described. 

13.  The  trigonometric  functions  of  a  (Fig.  8)  are  called 
the  sine^  cosine^  tangent^  cotangent^  secant^  and  cosecant  of  a. 
These  are  abbreviated  in  writing  to  sin  a,  cos  a,  tan  «,  cot  a, 
sec  a,  CSC  «,  and  are  defined  as  follows  : 

sin  a  =  P^^  =  ^,   whence  y  =  r  sin  a ; 
hyp.      r  ^  ' 

base       a?        i 

cos  a  =  i: —  =  ~9  whence  x  =  r  cos  a ; 
hyp.      r  ' 

tan  a  =  ^^—^  =  ->  whence   y  =  x  tan  a ; 
base      oc  ^  ' 

cot  a  = =  —J  whence  x  =  y  cot  a; 

perp.     y  ^  ' 

sec  a  =  —^  =  —9  whence   r  —  x  sec  a; 
base      a?  ' 

CSC  a  =  — ^  =  -9  whence   r  —  y  esc  a. 
perp.     y  ^ 

1  —  cos  a  and  1  —  sin  a,  called  versed-sine  a  and  coversed-sine  a,  respec- 
tively, are  sometimes  used. 

Ex.  1.  Write  the  trigonometric  functions  of  f3,  NPO  (Fig.  8),  and 
compare  with  those  of  a  above. 

The  meaning  of  the  prefix  co  in  cosine,  cotangent,  and  cosecant 
appears  from  the  relations  of  Ex.  1.  For  the  sine  of  an  angle  equals  the 
cosine,  i.e.  the  complement-sine,  of  the  complement  of  that  angle  ;  the  tangent 


THE   TRIGONOMETRIC   FUNCTIONS. 


11 


of  an  angle  equals  the  cotangent  of  its  complementary  angle,  and  the  secant 
of  an  angle  equals  the  cosecant  of  its  complement- 
ary angle. 

2.  Express  each  side  of  triangle  ABC  in 
terms  of  another  side,  and  some  function  of  an 
angle  in  all  possible  ways,  as  a  =  6  tan  A,  etc.  Fig.  10. 

14.  Constancy  of  the  trigonometric  functions.  It  is  iiiipor- 
taiit  to  notice  why  these  ratios  are  functions  of  the  angle,  i.e. 
are  the  same  for  equal  angles  and  different  for  unequal 
angles.     This  is  shown  by  the  principles  of  similar  triangles. 


\ 


Fig.  11. 

In  each  figure  show  that  in  all  possible  triangles  of  refer- 
ence for  a  the  ratios  are  the  same,  but  in  the  triangles  of 
reference  for  a  and  a',  respectively,  the  ratios  are  different. 

The  student  must  notice  that  sin  a  is  a  single  symbol.  It  is  the  name 
of  a  number,  or  fraction,  belonging  to  the  angle  a ;  and  if  it  be  at  any 
time  convenient,  we  may  denote  sin  «  by  a  single  letter,  such  as  o,  or  x. 
Also,  sin^a  is  an  abbreviation  for  (sin  «)'-^,  i.e.  for  (sin  a)  x  (sin  «). 
Such  abbreviations  are  used  because  they  are  convenient.  Lock,  Ele- 
mentary Trigonometry. 

15.  Fundamental  relations.  From  the  definitions  of  Art.  13 
the  following  reciprocal  relations  are  apparent : 


sin  a  = 


a  = 


tana 


CSC  a 

1 
sec  a' 

1 


cot  a 

Also  from  the  definitions. 


tana  = 


sm  g 
cos  a 


C8C  a 


sm  a 
1 


sec  a  = 1 

cos  a 

1 

cot  a 


cot  a  — 


tan  oL 


cos  a 
sin  a 


12  ^    PLANE   TRIGONOMETRY. 

From  the  right  triangle  NOP,  page  9, 
y'^  -\-  x^  =  T^  \ 


/2        ^2 


whence  (1)  U-j^'L^l^ 


From  (1)     sin^a+cos^  a=l;    sma=  Vl  — cos^  a;    cos  cc=? 

(2)  tan2a  +  l  =  sec2a;    ^^/^  «=  -y/sec^  cc—  1 ;     sec  a  =  ? 

(3)  l+cot2a  =  csc2a;     cota=Vcsc^  a—1 ;     esc  a  =  ? 

The  foregoing  definitions  and  fundamental  relations  are  of 
the  highest  importance,  and  must  he  mastered  at  once.  The 
student  of  trigonometry  is  helpless  without  perfect  familiarity 
with  them. 

These  relations  are  true  for  all  values  of  a,  positive  or  negative,  but 
the  signs  of  the  functions  are  not  in  all  cases  positive,  as  appears  from 
the  fact  that  in  the  triangles  of  reference  in  Fig.  8  x  and  y  are  sometimes 
negative.  The  equations  sin  a  =  ±  Vl  —  cos^  a,  tan  a=±  Vsec^  a—1, 
cot  a  =  ±  Vcsc^  ct  —  1,  have  the  double  sign  ± .  Which  sign  is  to  be  used 
in  a  given  case  depends  on  the  quadrant  in  which  a  lies. 

16.  The  relations  of  Art.  15  enable  us  to  express  any 
function  in  terms  of  any  other,  or  when  one  function  is 
given,  to  find  all  the  others. 

Ex.  1.   To  express  the  other  functions  in  terms  of  tangent : 


.inct-     ^     -           ^           -       ^^"^^       • 

CSC  a     VI  +  cot2  a      VH-  tan^  a 
1                  1 

tana 

sec  a  =  VI  +  tan2  a ; 

sec  a      Vl  +  tan2a 

tan  a  =  tan  a ; 

C8C«^^l+**»^«. 

tan  a 


THE   TRIGONOMETRIC   FUNCTIONS. 


13 


In  like  manner  determine  the  relations  to  complete  the  following 
table  ; 


sm  a 


cos  a 


tan« 


cot  a 


tan  a 


sm  a 

cos  a 
tana 
cot  a 
sec  a 
CSC  a 


VI  +  tan2  a 
1 


Vl  +  tan"'^  a 

tan  a 

1 
tan  a 

Vl  +  tan2  a 
Vl  +  tan2  a 


tan  a 


2.    Given  sin  a  =  f ;  find  the  other  functions. 


a=Vl  -^5  =  ^V7;   tan 


=  fV7; 


\V7      V7 

^          r-                14/-  14 

cot  a  =  —^  =  ^  V7 ;  sec  a  = =  —  =  f  V7 ;  esc  «  =  -  =  -• 


fV7 


iV7      V7 


3;   Given  tan  (f>  +  cot  ^  =  2 ;  find  sin  <^. 


tan  <f)  + 


tan  <f> 


2,   tan2  <^  -  2  tan  <^  +  1  =  0,   tan  <^  =  1. 


.♦.  sin  <f>  = 


tan  <^ 


Vl  +  tan2  <^ 


=  iV2. 


Or,  expressing  in  terms  of  sine  directly,  ?11L2_(_        ^  =  2, 

cos  <^     sin  <^ 

sin^  <^  +  cos^  (fi  =  2  sin  <^  cos  <fi,   sin^  <^  —  2  sin  ^  cos  <f>  +  cos^  ^  =  0 ; 
whence        sin  <^  —  cos  <^  =  0,   sin  <f>  =  cos  <f>.    .*.  sin  ^  =  ^  V2. 

4.  Prove  sec^  x  —  sec^  x  =  tan^  x  +  tan^  x. 

sec^x  —  sec^x  =  sec^  a:  (sec^  a:  —  1)  =  (1  +  tan^  x)  tan^  a:  =  tan^a:  +  tan*  a:. 

5.  Prove  sin^  y  +  cos®  ?/  =  1  —  3  sin^  y  cos^  y. 

sin®  y  +  cos®  y  =  (sin^  y  +  cos^  y)  (sin*  y  —  sin^  y  cos^  ?/  +  cos*  y) 

=  (sin^  2/  +  cos^  ?/)2  —  3  sin^  y  cos^  y  =  1  —3  sin^  ^  cos^  y. 


14  PLANE   TRIGONOMETRY. 


6.   Prove -i^2^  +  _22t^  =  sec.  CSC. +  1. 
1  —  cot  z     1  —  tan  z 

sing  cos  2 

tan  z  cot  z     _     cos  z  sin  2 


cot .     1  —  tan  z     I  _  cos  2 


COS. 


cos  .  (sin  .  —  COS  .)      sin  z  (cos  z  —  sin  z) 

_  sin^  .  —  cos^  .  _  sin^  .  +  sin  .  cos  z  +  cos^  . 

sin  .  cos  .  (sin  .  —  cos  z)  sin  z  cos  . 

1  +  sin .  cos .  1.1  ,1 

=  —^. = h  1  =  sec  .  esc  2  +  1. 

sm  .  cos  .         sin  z  cos  z 

In  solving  problems  like  3,  4,  5,  and  6  above,  it  is  usually  safe,  if  no 
other  step  suggests  itself,  to  express  all  other  functions  of  one  member 
in  terms  of  sine  and  cosine.  The  resulting  expression  may  then  be  re- 
duced by  the  principles  of  algebra  to  the  expression  in  the  other  member 
of  the  equation.  For  further  suggestions  as  to  the  solution  of  trigono- 
metric equations  and  identities  see  page  66. 

EXAMPLES. 

1.  Find  the  values  of  all  the  functions  of  a,  if  sin  a  =  | ;  if  tan  a  =  f ; 
if  sec  r}t  =  2 ;  if  cos  a  =  ^V3 ;  if  cot  a  =  | ;  if  esc  ot  =  V2. 

2.  Compute  the  functions  of  each  acute  angle  in  the  right  triangles 

whose  sides  are :  (1)  3,  4,  5;   (2)  8,  15,  17;   (3)  480,  31,  481 ;   (4)  a,b,c; 

yr-^    2  xy      x^  +  y^ 

(5)  ^,        ^  ^  ,  x+y. 

X  —  y      X  —  y 

3.  If  cos  a  =  j\,  find  the  value  of  si^<^  +  ^^^^^. 

cos  a  — cot  a 

4.  If  2  cos  a  =  2  —  sin  ct,  find  tan  a. 

5.  If  sec^  a  csc^  a  —  4  =  0,  find  cot  a. 

6.  Solve  for  sin  ^  in      13  sin  /?  +  5  cos^  ^  =  11. 
Prove 

7.  sin*  <;^  —  cos*  <^  =  1  —  2  cos^  <^. 

8.  (sin  a  +  cos  a)  (sin  a  —  cos  a)  =  2  sin^  a  —  1. 

9.  (sec  a  +  tan  a)  (sec  a  —  tan  a)  =  1. 

10.   cos2  y8  (sec2 13-2  sin2  ^)  =  cos*  jS  +  sin*  (3. 
cos  V 


11.   tan  V  +  sect' 
12. 


1  —  sin  V 
sin  w         1  +  cos  w 


1  —  cos  w         sin  w 
13.    (sec^  +  l)(l-cos^)  =  tan2^cosA 


FUNCTIONS  OF   CERTAIN   ANGLES.  15 

14.  sin*  t  —  siii2 1  =  cos*  t  —  cos^  t. 

15.  -ilH^  + 1+^  =  sec^^ (CSC fi  +  1). 
1  —  Sin  y8         smfi 

16.  (tan  A  +  cot  Ay  =  sec2 ^  csc^  ^. 

17.  sec^  ar  —  sin^  a;  =  tan^  a:  +  cos^  x. 

In  the  triangle  ABC,  right  angled  at  C, 

18.  Given  cos  A  =  ^y  BC  =  45,  find  tan  B,  and  AB. 

19.  If  cos  A  =  ^l  ~  ""l  and  AB  =  m^  +  n%  find  ^  C  and  ^C. 

20.  If  ^  C  =  m  +  n,  £C  =  m  —  n,  find  sin  A,  cos  5. 

21.  In    examples    18,    19,    20,    above,    prove    sin^  ^4  4- cos^  .4  =  1 ; 

1  +  tan2  A  =  sec2  A . 

17.  Functions  of  certain  angles.  The  trigonometric  func- 
tions are  numerical  quantities  which  may  be  determined  for 
any  angle.  In  general  these  values  are  taken  from  tables 
prepared  for  the  purpose,  but  the  principles  already  studied 
enable  us  to  calculate  the  functions  of  the  following  angles. 

18.  Functions  of  O''.  If  a  be  a  very  small  angle,  the 
value  of  y  is  very  small,  and 
decreases  as  a  diminishes. 
Clearly,  when  a  approaches 
0°  as  a  limit,  ^  likewise  ap- 
proaches 0,  and  X  approaches  r,  so  that  when  a  =  0°, 

^  =  0,  and  X  =  r. 

.-.  «mO°  =  ^  =  0,  co^ 0°  =  — i— =  QO, 

r 

r 

tanO""  =  ^  =  0,  C8C  0°  =  -r^  =  00. 

X  sin  0° 

In  the  figure  of  Art.  18,  by  diminishing  a  it  is  clear  that  we  can  make 
y  as  small  as  we  please,  and  by  making  a  small  enough,  we  can  make  the 
value  of  y  less  than  any  assignable  quantity,  hoivever  small,  so  that  sin  a  ap- 
proaches as  a  limit  0.  This  is  what  we  mean  when  we  say  sin  0°  =  0. 
In  like  manner,  it  is  evident  that,  by  sufficiently  diminishing  a  we  can 
make  cot  a  greater  than  any  assignable  quantity.  This  we  express  by 
saying  cotO°  =  co. 


€OtO° 

± 

tanO° 

8ecO° 

1 

cos  0° 

nRn  0° 

1 

16 


PLANE  TRIGONOMETRY. 


19.   Functions  of  30°.     Let  NOP  be  the  triangle  of  refer- 
,22  ence  for  an  angle  of  30°.     Make 

triangle  NOP'  =  NOP.  Then 
POP'  is  an  equilateral  triangle 
(why?),  and  ON  bisects  PP'. 
Hence 

Also  X  =  Vr^  —  y^  =  V3^—  y V3. 
c%G  30°  =  2, 


Fig.  13 

nn  30°  =  ^  =  ^  =  ^' 
r      2^     2 


r       2y       ^ 


^an  30° 


y 


=  -4^=iV3, 


«/V3      V3 


se<?  30°  =  I V3, 
co^30°=V3. 


20.  Functions  of  45°.     Let  NOP  be  the  triangle  of  refer- 
ence.    If  angle  NOP  =  45°,  OPN^  45°. 


Then 


y  =  x^  and  r  =  Va^^  +  3/^  =  V2  a;^  =  a; V2. 


tn  45°  =  ^- 


sm 


V2 


iV2, 


cos  45' 


a:         a; 


-^=i-V2, 


^      a:V2 


Find  cot  45°,  sec  45°,  esc  45° 


ia^i  45°  =  ^  =  -  =  1. 

.T         X 


FUNCTIONS  OF   CERTAIN  ANGLES. 


17 


21.  Functions  of  60°.  The  functions  of  60°  may  be  com- 
puted by  means  of  the  figure,  or 
they  may  be  written  from  the  func- 
tions of  the  complement,  or  30°. 
Let  tlie  student  in  both  ways  show 
that 


sm60°=iV3,      cos  60°  =1 

tan  60°  =  Vs. 
Compute  also  the  other  functions  of  60°. 


Fig.  15. 


22.  Functions  of  90°.  If  «  be  an  angle  very  near  90°, 
the  value  of  x  is  very  small,  and  de- 
creases as  a  increases  toward  90°. 
Clearly  when  a  approaches  90°  as  a 
limit,  X  approaches  0,  and  ?/  ap- 
proaches r,  so  that  when 

"^¥  ^  a  =  90°,    x=0,    y  =  r. 

,  •.  sin  90°  =  1,  cos  90°  =  0,  tan  90°  =  oo . 


Fig.  16. 


Compute  the  other  functions.  Also  find  the  functions  of 
90°  from  those  of  its  complement,  0°. 

23.  It  is  of  great  convenience  to  the  student  to  remember 
the  functions  of  these  angles.  They  are  easily  found  by 
recalling  the  relative  values  of  the  sides  of  the  triangles  of 
reference  for  the  respective  angles^  or  the  values  of  the  other 
functions  may  readily  be  computed  by  means  of  the  funda- 
mental relations,  if  the  values  of  the  sine  and  cosine  are 
remembered,  as  follows  : 


a 

0° 

30° 

45° 

60° 

90° 

sine 
cosine 

iVo 

K/4 

1  rz 

2  Vo 

iV2 

W2 

iVO 

18  PLANE  TRIGONOMETRY. 

ORAL  WORK. 

1.  Which  is  greater,  sin  45°  or  I  sin  90=?  sin  60°  or  2  sin  30°? 

2.  ^  From  the  functions  of  60°,  find  those  of  30° ;  from  the  functions  of 
90°,  those  of  0°.  Why  are  the  functions  of  45°  equal  to  the  co-functions 
of  45°? 

3.  Given  sin  A  =  |,  find  cos  A  ;  tan  A. 

4.  Show  that  sin  B  esc  ^  =  1 ;  cos  C  sec  C  =  1 ;  cot  x  tan  x  =  1. 

5.  Show  that  sec2  0  -  tan^  0  =  csc2  0  -  cot^  6  =  sin2  $  +  cos^  6. 

6.  Show  that  tan  30°  tan  60°  =  cot  60°  cot  30°  =  tan  45°. 

7.  Showthattan60°sin2  45°  =  cos30°sin90°. 

8.  Show  that  cos  a  tan  a  =  sin  a ;  sin  ^  cot  /3  =  cos  13. 

9.  Show  that  1  -tan2  30°  ^  ^^^  g^^  ^  ^  ^os  0°. 

l-|-tan2  30°  ^ 

10.   Show  that  (tan  y  +  coty)  sin  y  cos  y  =  1. 

EXAMPLES. 

1.  Show  that  sin  30°  cos  60°  +  cos  30°  sin  60°  =  sin  90°. 

2.  Show  that  cos  60°  cos  30°  +  sin  60°  sin  30°  =  cos  30°. 

3.  Show  that  sin  45°  cos  0°  -  cos  45°  sin  0°  =  cos  45°. 

4.  Show  that  cos2  45°  -  sinHS"  =  cos  90°. 

5.  Show  that    tan  45°  + tan  0°   ^  ^^^^^o. 

1  -  tan  45°  tan  0° 

IiA=  60°,  verify 


i^-i- 


6.   sin^^         ,^-cos^ 


7.   tan"^ 


_    /l  —  cos  A 
2  ~  >'l  +  cos^' 

8.  cos^  =  2cos2^-l  =  l-2sin2:^. 

2  2 

licc  =  0°,l3  =  30°,  y  =  45°,  8  =  60°,  e  =  90°,  find  the  values  of 

9.  sin  13  +  cos  8. 

10.  cos  y8  +  tan  8. 

11.  sin  ^  cos  S  +  cos  ;8  sin  8  —  sin  e. 

12.  (sin  13  +  sin  e)  (cos  a  +  cos  8)  —  4  sin  a  (cos  y  +  sin  e) . 


VARIATIONS  IN   THE   FUNCTIONS. 


19 


24.   Variations  in  the  trigonometric  functions. 

Signs.  Thus  far  no  account  has  been  taken  of  the  signs  of 
the  functions.  By  the  definitions  it  appears  that  these  de- 
pend on  the  signs  of  a;,  ?/,  and  r.  Now  r  is  always  positive, 
and  from  the  figures  it  is  seen  that  x  is  positive  in  the  first 


8&H.  + 
Csc.  + 


X- 


(X-) 


(r+) 


Cot.  + 


Sin. 
Cos. 
Tan. 
Cot. 
Sec. 
Csc. 


\y-) 


Cos.  + 


Fia.  17. 


and  fourth  quadrants,  and  ^  is  positive  in  the  first  and 
second.     Hence 

For  an  angle  in  the  first  quadrant  all  functions  are  positive^ 
since  a:,  ^,  r  are  positive. 

In  the  second  quadrant  x  alone  is  negative.,  so  that  those 
functions  whose  ratios  involve  a:,  viz.  cosine.,  tangent^  co- 
tangent^ secant.,  are  negative;  the  others,  sine  and  cosecant., 
are  positive. 

In  the  third  quadrant  x  and  y  are  both  negative.,  so  that 
those  functions  involving  r,  viz.  sine.,  cosine.,  secant.,  cosecant., 
are  negative  ;  the  others,  tangent  and  cotangent.,  ?iVQ  positive. 

In  the  fourth  quadrant  y  is  negative^  so  that  sine^  tangent., 
cotangent,  cosecant  are  negative.,  and  cosine  and  secant.,  positive. 

Values.  In  the  triangle  of  reference  of  any  angle,  the 
hypotenuse  r  is  never  less  than  x  or  y.  Then  if  r  be  taken  of 
any  fixed  length,  as  the  angle  varies,  the  base  and  perpen- 
dicular of  the  triangle  of  reference  may  each  vary  in  length 

X  11 

from  0  to  r.    Hence  the  ratios  -  and  -  can  never  be  greater 

r  r  ° 

r  r 
than  1,  nor  if  x  and  y  are  negative,  less  than  —1;  and  — >  - 

X  y 


20 


PLANE   TRIGONOMETRY. 


cannot  have  values  between  +  1   and  —  1.      But  the  ratios 

^  and  -  may  vary  without  limit,  i.e.   from    +  oo    to  —  oo. 
X  y 

Therefore  the  possible  values  of  the  functions  of  an  angle 

are  : 

sine  and  cosine  between  +  1  and  —  1, 

i.e.  sine  and  cosine  cannot  he  numerically  greater  than  1; 

tangent  and  cotangent  between  +  oo  and  —  oo, 

i.e.  tangent  and  cotangent  may  have  any  real  value  ; 

secant  and  cosecant  between  +  oo  and  +  1,  and  —  1  and  —  oo, 

i.e.  secant  and  cosecant  may  have  any  real  values.,  except 
values  between  +  1  and  —  1. 

These  limits  are  indicated  in  the  following  figures.     The 
student  should  carefully  verify. 


Sin.  + 1 
Cos.  -  0 
Tan.   —00 


90° 
Y 


\Sin  0=±0        o  /. 
-X      180  X- 


1,4  0 


Sin. 
Cos. 
Tan. 


-1 
-0 

+  00 


+  1 
+  0 

+00 


X 


0,   +1,-0    360 


-1 
+  0 


F' 


370° 


Fig.  18. 


25.  In  tracing  the  changes  in  the  values  of  the  functions  as 
a  changes  from  0°  to  360°,  consider  the  revolving  line  r  as 
of  fixed  length.  Then  x  and  y  may  have  any  length  between 
0  and  r. 


y 


0 


Sine.    At  0°,  sin  «="=-  =  0.     As  a  increases  through 

r      r 

y  ^ 

the  first  quadrant,  y  increases  from  0  to  r,  whence  -  increases 
from  0  to  1.     In  passing  to  180°  sin  a  decreases  from  1  to  0, 


VARIATIONS  IN   THE   FUNCTIONS.  21 

since  y  decreases  from  r  to  0.  As  «  passes  through  180°,  y 
changes  sign,  and  in  the  third  quadrant  decreases  to  nega- 
tive r,  so  that  sin  a.  decreases  from  0  to  —  1.  In  the  fourth 
quadrant  y  increases  from  negative  r  to  0,  and  hence  sin  a 
increases  from  —  1  to  0. 

Cosine  depends  on  changing  values  of  x.  Show  that, 
as  a  increases  from  0°  to  360°,  cos  «  varies  in  the  four 
quadrants  as  follows:  1  to  0,  0  to  —  1,  —  1  to  0,  0  to  1. 

Tangent  depends  on  changing  values  of  both  y  and  x. 

At    0°,  ^  =  0,  a:  =  r,  at  180°,  y  =  0,x  =  -r, 

at  90°,  x  =  0,y  =  r,  at  270°,  x=0,y  =  -r, 

V      0 
Hence  tan  0°  =  -^  =  -  =  0.     As  a  passes  to  90°,  y  increases 
X      r 

to  r,  and  x  decreases  to  0,  so  that  tan  a  increases  from  0  to  oo. 
As  a  passes  through  90°,  x  changes  sign,  so  that  tan  a 
changes  from  positive  to  negative  by  passing  through  oo. 
In  the  second  quadrant  x  decreases  to  negative  r,  y  to  0,  and 
tan  a  passes  from  —  oo  to  0.  As  a  passes  through  180°, 
tana  changes  from  minus  to  plus  by  passing  through  0, 
because  at  180°  y  changes  to  minus.  In  the  third  quadrant 
tana  passes  from  0  to  oo,  changing  sign  at  270°  by  passing 
through  00,  because  at  270°  x  changes  to  plus.  In  the  fourth 
quadrant  tan  a  passes  from  —  oo  to  0. 

Cotangent.  In  like  manner  show  that  cot  a  passes  through 
the  values  oo  to  0,  0  to  —  oo,  oo  to  0,  0  to  —  oo,  as  a  passes 
from  0°  to  360°. 

Secant  depends  on  x  for  its  value.  Noting  the  change 
in  X  as  under  cosine,  we  see  that  secant  passes  from  1  to  oo, 

—  oo  to  —  1,  —  1  to  —  00,  00  to  1. 

Cosecant  passes   through   the   values   oo    to    1,   1   to   oo, 

—  00    to    —  1,    —  1    to    —  00. 

The  student  should  trace  the  changes  in  each  function 
fully,  as  has  been  done  for  sine  and  tangent,  giving  the 
reasons  at  each  step. 


22 


PLANE   TRIGONOMETRY. 


a 

0°  to  90° 

90°  to  180° 

180°  to  270° 

270°  to  360° 

sin 

0  to  1 

1  to  0 

-  0  to  -  1 

-  1  to  -  0 

cos 

1  to  0 

-  0  to  - 1 

-1  to  -0 

0  to  1 

tan 

0  to  00 

-  00  to  -  0 

0  to  00 

-  00  to  -  0 

cot 

00  to  0 

-  0  to  -  00 

00  to  0 

-  0  to  -  00 

sec 

1    to    GO 

—  00   to  —  1 

—  1   to    —  00 

00  to  1 

CSC 

00    to    1 

1    to    00 

—  00  to  —  1 

—    1    to    —00 

*  26.  Graphic  representation  of  functions.  These  variations 
are  clearly  brought  out  by  graphic  representations  of  the 
functions.  Two  cases  will  be  considered  :  I,  when  a  is  a 
constant  angle  ;  II,  when  a  is  a  variable  angle. 

I.    When  a  is  a  constant  angle. 

The  trigonometric  functions  are  ratios,  pure  numbers. 
By  so  choosing  the  triangle  of  reference  that  the  denomi- 
nator of  the  ratio  is  a  side  of  unit  length,  the  side  forming 
the  numerator  of  that  ratio  will  be  a  geometrical  representa- 
tion of  the  value  of  that  function,  e.g.  if  in  Fig.  19  r  =  1, 

then  sin  a  =  ?  =  ^=^.     This  may  be  done  by  making  a  a 

central  angle  in  a  circle  of  radius  1,  and  drawing  triangles 
of  reference  as  follows : 


Fio.  19. 


GRAPHIC   REPRESENTATION   OF   FUNCTIONS.  23 

In  all  the  figures  A  OF  =  a,  and 

BP       BP        j.r> 


OB      OB     ^j, 

BP     AD     AD       .J, 
''""'^OB^OA^    1    =^^' 

OA     BO     EC      T,n 

OP      OB      OB      r.j. 
'''''- OB=OA=    1    =^^' 

OP      00      00     rtn 

It  appears  then  that,  by  taking  a  radius  1, 

sine  is  represented  by  the  perpendicular  to  the  initial  line, 
drawn  from  that  line  to  the  terminus  of  the  arc  sub- 
tending the  given  angle; 

cosine  is  represented  by  the  line  from  the  vertex  of  the 
angle  to  the  foot  of  the  sine ; 

tangent  is  represented  by  the  geometrical  tangent  drawn 
from  the  origin  of  the  arc  to  the  terminal  line,  produced 
if  necessary; 

cotangent  is  represented  by  the  geometrical  tangent  drawn 
from  a  point  90°  from  the  origin  of  the  arc  to  the 
terminal  line,  produced  if  necessary; 

secant  is  represented  by  the  terminal  line,  or  the  terminal 
line  produced,  from  the  origin  to  its  intersection  with 
the  tangent  line ; 

cosecant  is  represented  by  the  terminal  line,  or  the  terminal 
line  produced,  from  the  origin  to  its  intersection  with 
the  cotangent  line. 


24 


PLANE   TRIGONOMETRY. 


These  lines  are  not  the  functions^  but  in  triangles  drawn 
as  explained  their  lengths  are  equal  to  the  numerical  values 
of  the  functions,  and  in  this  sense  the  lines  may  be  said  to 
represent  the  functions.  It  will  be  noticed  also  that  their 
directions  indicate  the  signs  of  the  functions.  Let  the 
student  by  means  of  these  representations  verify  the  results 
of  Arts.  24  and  25. 


II.    When  a  is  a  variable  angle. 

Take  XX'  and  YY'  as  axes  of  reference,  and  let  angle 
units  be  measured  along  the  ic-axis,  and  values  of  the  func- 
tions parallel  to  the  ?/-axis,  as  in  Art.  3.  We  may  write 
corresponding  values  of  the  angle  and  the  functions  thus : 

a=0°,  30°,  45%  60°,  90°,  120°,  135°,  150°,  180°,  210°,  225°, 
sin«=0,  i,  iV2,  iV3,  1,  iV3,  iV2,  i,   0,   _  i,  _  i  V2, 

a=  240°,  270°,  300°,  315°,  330°,  360°,  -30°,  -45°,  -60°,  -90°,  etc., 
sina=-|V3,  -1,  -iV3,  -^^2,  -\,     0,   -^  -iV2,  -|V3,  -1,  etc. 

These  values  will  be  sufficient  to  determine  the  form  of  the 
curve  representing  the  function.     By  taking  angles  between 

those  above,  and   computing 
the  values  of  the  function,  as 
given  in  mathematical  tables, 
the  form  of  the  curve  can  be 
/^  determined   to   any  required 
degree  of  accuracy.     Reduc- 
ing   the    above    fractions    to 
decimals,  it  will  be  convenient 
to  make  the  ?/-units  large  in 
comparison  with  the  a:-units. 
In  the  figure  one  a^-unit  repre- 
sents 15°-,  and  one  y-unit  0.25. 
Measuring  the  angle  values  along  the  rr-axis,  and  from  these 
points  of  division  measuring  the  corresponding  values  of  sin  a 
parallel  to  the  ^-axis,  as  in  Art.  3,  we  have,  approximately. 


Curves  of  Sine  and  Cosecant. 


Cosecant 

Fig.  20. 


GRAPHIC   REPKESENTATION   OF  FUNCTIONS.  25 

OX^  =  30°  =  2  units,  OX^  =  45°    =3  units, 

Xi  Fi  =  1      =2  units,        Xg  Fg  =  0. 71  =  2. 84  units, 

OXg  =60°    =4  units,  etc., 
X^Y^=0.S6  =  3.44  units,  etc. 

We  have  now  only  to  draw  through  the  points  F^,  Fg,  Fg, 
etc.,  thus  determined,  a  continuous  curve,  and  we  have  the 
sine-curve  or  sinusoid. 

The  dotted  curve  in  the  figure  is  the  cosecant  curve.  Let 
the  student  compute  values,  as  above,  and  draw  the  curve. 

In  like  manner  draw  the  cosine  and  secant  curves,  as 
follows  : 

i 


Curves  of  Cosine  and  Secant, 

Cosine 

Secant ~ 

FiG.  21. 


Tangent  curve.     Compute  values  for  the  angle  a  and  for 
tan  a,  as  before  ; 


a  =  0°,  30°,  45°,  60°,  90°,  120°,  136°,  150°,  180°,  210°,  226°,  240°,  270°, 
tan  a  =  0,  \VS,  1,    V3,  ±oo,  -  V3,  -1,  -^VS,  0,    |\/3,    1,     V3,  ±oo, 

a  =  -  30°,  -  45°,  -  60°,  -  90°,  etc., 
tan  a  =  -  ^  V3,  -  1,    -  V3,    ±  oo,  etc. 

Then  lay  off  the  values  of  a  and  of  tan  a  along  the  x^  and 
parallel  to  the  ?/-axis,  respectively.     It  will  be  noted  that, 


26 


PLANE  TRIGONOMETRY. 


as  a  approaches  90°,  tan  a  increases  to  oo,  and  when  a  passes 
90°,  tan  a  is  negative.    Hence  the  value  is  measured  parallel 


Curves  of  Tangent  and  Cotangent. 

Tangent    

Cotangent 

Fig.  22.      ' 


to  the  ^-axis  downward,  thus  giving  a  discontinuous  curve, 
as  in  the  figure. 


27.   The  following  principles  are  illustrated  by  the  curves  : 

1.  The  sine  and  cosine  are  continuous  for  varying  values 
of  the  angle,  and  lie  within  the  limits  +  1  and  —  1.  Sine 
changes  sign  as  the  angle  passes  through  180°,  360°,  ••♦, 
n  180°,  while  cosine  changes  sign  as  the  angle  passes  through 
90°,  270°,  •••,  (2  7^  +  1)  90°.  Tangent  and  cotangent  are 
discontinuous,  the  one  as  the  angle  approaches  90°,  270°,  •••, 
(2?i  +  l)  90°,  the  other  as  the  angle  approaches  180°,  360°,  •-, 
7il80°,  and  each  changes  sign  as  the  angle  passes  through 
these  values.  The  limiting  values  of  tangent  and  cotangent 
are  +  oo  and  —  oo. 

2.  A  line  parallel  to  the  ?/-axis  cuts  any  of  the  curves  in 
but  one  point,  showing  that  for  any  value  of  a  there  is  but 
one  value  of  any  function  of  a.  But  a  line  parallel  to  the 
a?-axis  cuts  any  of  the  curves  in  an  indefinite  number  of 
points,  if  at  all,  showing  that  for  any  value  of  the  function 
there  are  an  indefinite  number  of  values,  if  any,  of  a. 


GRAPHIC   REPRESENTATION  OF  FUNCTIONS.  27 

3.  The  carves  afford  an  excellent  illustration  of  the  varia- 
tions in  sign  and  value  of  the  functions,  as  a  varies  from  0 
to  360°,  as  discussed  in  Art.  25.  Let  the  student  trace  these 
changes. 

4.  From  the  curves  it  is  evident  that  the  functions  are 
periodic^  i.e.  each  increase  of  the  angle  through  360°  in  the 
case  of  the  sine  and  cosine,  or  through  180°  in  the  case  of 
the  tangent  and  cotangent,  produces  a  portion  of  the  curve 
like  that  produced  by  the  first  variation  of  the  angle  within 
those  limits. 

5.  The  difference  in  rapidity  of  change  of  the  functions 
at  different  values  of  a  is  important,  and  reference  will  be 
made  to  this  in  computations  of  triangles.  (^See  Art.  64, 
Case  III.)  A  glance  at  the  curves  shows  that  sine  is  chang- 
ing in  value  rapidly  at  0°,  180°,  etc.,  while  near  90°,  270°, 
etc.,  the  rate  of  change  is  slow.  But  cosine  has  a  slow  rate 
of  change  at  0°,  180°,  etc.,  and  a  rapid  rate  at  90°,  270°,  etc. 
Tangent  and  cotangent  change  rapidly  throughout. 

Ex.    Let  the  student  discuss  secant  and  cosecant  curves. 

ORAL  WORK. 

1.  Express  in  radians  180°,  120°,  45°;  in  degrees,  J^  radians,  2  it, 
Itt,  f  tt. 

2.  If  ^  of  a  right  angle  be  the  unit,  what  is  the  measure  of  |  of  a 
right  angle?  of  90°?  of  135°? 

3.  Which  is  greater,  cos  30°  or  I  cos  60°?  tan  -  or  cot-?  sin  ^  or  cos-  ? 

^  ^  6  3  4  4 

4.  Express  sin  a  in  terms  of  sec  a ;  of  tan  a ;  tan  a  in  terms  of  cos  a ; 
of  sec  a. 

5.  Given  sin  a  =  |,  find  tan  a.  If  tan  cc  =  1,  find  sin  a,  esc  a,  cot  a ; 
also  tan  2  a,  sin  2  a,  cos  2  a. 

6.  If  cos  a  =  i,  find  sin  -,  tan  -• 

2  2 

7.  In  what  quadrant  is  angle  t,  if  both  sin  t  and  cos  t  are  minus  ?  if 
sin  t  is  plus  and  cos  t  minus  ?  if  tan  t  and  cot  t  are  both  minus  ?  if  sin  < 
and  CSC  t  are  of  the  same  sign  ?    Why  ? 

8.  Of  the  numbers  3,  ^,  —  5,  —  4,  a,  —  6,  oo,  0,  which  may  be  a  value 
of  sin  JO  ?  of  sec  j9  ?  of  tan  p  ?     Why  ? 


28  PLANE   TKIGONOMETRY. 


EXAMPLES. 

1.  If  sin  26°  40'  =  0.44880,  find,  correct  to  0.00001,  the  cosine  and 
tangent. 

2.  If  tan  a  =  VS,  and  cot  fi  =  |  Vs,  find  sin  a  cos  ^  —  cos  a  sin  /i. 

3     Evaluate  si"  ^0°  cot  30°  -  cos  eO'^  tan  60° 
sin  90°  cos  0° 

Prove  the  identities : 

4.  tan^(l  -cot2^)  +  cot^(l  -tan2^)  =  0. 

5.  (sin^  +  sec^)2  +(cos^  +  csc^)2  =(1  +  sec^  csc^)^. 

6.  sin2  X  cos  a:  esc  a;  —  cos^  x  esc  x  sin^  x  +  cos*  x  sec  a:  sin  a?  =  sin^  x  cos  x 
4-  cos^  X  sin  x. 

7 .  tan^  w  +  cot^  w  =  sec^  w  csc^  w  —  2. 

8.  sec^  V  +  cos2  V  =  2  -{-  tan^  v  sin^  v. 

9.  cos2«  +  1  =  2cos3«sec^ +  sin2^ 

10.  csc2 1  —  sec2 1  =  cos2 « csc^  t  -  sin2  <  sec^  t. 

11.  The  sine  of  an  angle  is  ^„  ~  ^i  find  the  other  functions. 

12.  If  tan^  +  sin  J.  =  m,  tan  ^  —  sin  ^  =  n,  prove  m'^  —  n^  =  4:Vmn. 

Solve  for  one  function  of  the  angle  involved  the  equations : 

13.  sin^  + 2cos^  =  1.  16.  2sin2ar  +  cosa;  -  1  =  0. 
cosa_3  17.  sec^a;  —  7tana:  —  9  =  0. 
tana     2                                          18.   3  cscy  +  lOcoty  -  35  =  0. 

15.    \/3csc2^  =  4cot^.  19.   sin^i; -|cosu- 1  =  0. 

20.  a  sec^  zo  +  b  tan  w  +  c  —  a  =  0. 

21.  K  ^HLd  =  V2,  *HLd  =  V3,  find  A  and  5. 

sin  ^  tan  B 

22.  Find  to  five  decimal  places  the  arc  which  subtends  the  angle  of 
1°  at  the  centre  of  a  circle  whose  radius  is  4000  miles. 

23.  If  CSC  A  =  f  V3,  find  the  other  functions,  when  A  lies  between 
—  and  TT. 

24.  In  each  of  two  triangles  the  angles  are  in  G.  P.  The  least  angle 
of  one  of  them  is  three  times  the  least  angle  of  the  other,  and  the  sum  of 
the  greatest  angles  is  240°.  Find  the  circular  measure  of  each  of  the 
angles. 


CHAPTER  III. 

FUNCTIONS  or  ANY  ANGLE  —  INVERSE  FUNCTIONS. 

28.  By  an  examination  of  the  figure  of  Art.  24  it  is  seen 
that  all  the  fundamental  relations  between  the  functions  hold 
true  for  any  value  of  a.  The  table  of  Art.  16  expresses  the 
functions  of  a,  whatever  be  its  magnitude,  in  terms  of  each 
of  the  other  functions  of  that  angle  if  the  ±  sign  be  prefixed 
to*  the  radicals. 

The  definitions  of  the  trigonometric  functions  (Art.  12) 
apply  to  angles  of  any  size  and  sign,  but  it  is  always  possible 
to  express  the  functions  of  any  angle  in  terms  of  the  func- 
tions of  a  positive  acute  angle. 

The  functions  of  any  angle  ^,  greater  than  360°,  are  the 
same  as  those  of  ^  ±  w  •  360°,  since  6  and  6  ±n  •  360°  have 
the  same  triangle  of  reference.  Thus  the  functions  of  390°, 
or  of  750°,  are  the  same  as  the  functions  of  390°  —  360°,  or 
of  750°—  2-360°,  i.e.  of  30°,  as  is  at  once  seen  by  drawing  a 
figure.  So  also  the  functions  of  —315°,  or  of  —675°  are 
the  same  as  those  of  -  315°  +  360°,  or  of  -  675°  +  2-360°, 
i.e.  of  45°. 

For  functions  of  angles  less  than  360°  the  relations  of  this 
chapter  are  important. 

29.  To  find  the  relations  of  the  functions  of  —  ^,  90°  ±  ^, 
180°  ±  6,  and  270°  ±6  to  the  functions  of  6,  6  being  any  angle. 

Four  sets  of  figures  are  drawn,  I  for  d  an  acute  angle,  II 
for  Q  obtuse.  III  for  0  an  angle  of  the  third  quadrant,  and 
IV  for  d  an  angle  of  the  fourth  quadrant. 

In  every  case  generate  the  angles  forming  the  compound 
angles  separately,  i.e.  turn  the  revolving  line  first  through 


30  PLANE  TRIGONOMETRY, 

(a)  (6) 


(c) 


x' 


^r' 


III 


in 


III 


IV 


IV 
Fig.  23. 


t 

\r' 

y 

Y\ 

\ 

oW      ] 

rCA 

y 

M~ 

1%'i'' 

IV 


FUNCTIONS  OF   ANY  ANGLE. 


31 


/ 

\ 

Ix' 

1^ 

\  / 

\  -f 

t 

V 

\#/ 

y 

V" 

N 

y 

K^. 

// 

,  \?.io 

—  fl  / 

y 

\ 

y 

x\ 

c^y 

*>0 

+8 

II 


II 


III 


t 

[V 

ry^ 

y 

/.X 

> 

<^'y' 

vs= 

< 

y 

»0 

=9 

IV 


^ 

+^ 

a;'A/1 

N\  X' 

I  v 

f 

y 

/ 

y 

/f/ 

'''\ 

/ 

\ 

IV 


Fig.  23. 


32  PLANE  TRIGONOMETRY. 

0°,  90°,  180°,  or  270°,  and  then  from  this  position  through 
^,  or  —  ^,  as  the  case  may  be.  Form  the  triangles  of  refer- 
ence for  (a)  the  angle  (9,  (6)  -  ^,  (c)  180°  ±  (9,  (^d)  90°  ±  (9, 
(e)  270°  ±^. 

The  triangles  of  reference  (a),  (6),  (<?),  (c?),  and  (e),  in 
each  of  the  four  sets  of  figures,  I,  II,  III,  IV,  are  similar, 
being  mutually  equiangular,  since  all  have  a  right  angle  and 
one  acute  angle  equal  each  to  each.  Hence  the  sides  x^  y^  r 
of  the  triangles  (a)  are  homologous  to  x\  y\  r'  of  the  cor- 
responding triangles  (5)  and  (c),  but  to  ?/',  x\  7•^  of  the 
corresponding  triangles  (c?)  and  {e).  For  the  sides  x  of 
triangle  {a)  and  x^  of  the  triangles  (6)  and  (c)  are  opposite 
equal  angles,  and  hence  are  homologous,  but  the  sides  y^  are 
opposite  this  same  angle  in  triangles  (c?)  and  (e),  and  there- 
fore sides  y^  of  (c?)  and  (e)  are  homologous  to  x  of  (a). 

Attending  to  the  signs  of  x  and  a;',  y  and  ?/'  in  the  similar 
triangles  {a)  and  (6), 

sin(-^)=|^  =  -|=-sin^, 
cos(-6>)=^  =  ^      =cos(9, 

tan (-  6>)  =  ^=  - ^  =  -  tan^. 

^      a?'  X 

Also  in  the  similar  triangles  {a)  and  (c), 

sin  (180°  -  (9)  =  ^  =  ^     =  sin  (9, 


r 


= =  —  cos  r, 


cos  (180°-^)  =  =^ 

tan  (180°  -  ^)=  ^  =  -  ^  =  -  tan(9, 
In  like  manner  show  that 

sin  (180°  + (9)  = -sin  (9, 
cos  (180°  + (9)  =  - cos  ^, 
tan  (180°  +  ^)=  tan  ^. 


FUNCTIONS  OF  ANY  ANGLE.  33 

Again,  in  the  similar  triangles  (a)  and  ((^), 

sin  (90°  +  (9)  =  ^  =  -     =  cos^, 

cos  (90°  4-  ^)  =  ^  =  -  -  =  -  sin  ^, 


r 
.f 


tan  (90°  +  (9)  =  ^  =  -  -=  -  cot  ^. 
Show  that 

sin  (90°-^)=  cos  ^,  C 

cos(90°-6»)=sin(9,  |     ^ 

tan  (90° -(9)  =  cot  (9. 

Finally,  from  the  similar  triangles  (a)  and  (e),  show  that 

sin  (270°  ±  (9)=- cos  6^, 

cos(270°±^)=±sin^, 

tan  (270°  ±^)=Tcot^. 

From  the  reciprocal  relations  the  student  can  at  once 
write  the  corresponding  relations  for  secant,  cosecant,  and 
cotangent. 

30.  Since  in  each  of  the  four  cases  x\  y'  of  triangles 
(6)  and  (<?)  are  homologous  to  x^  y  of  triangle  (a)^  while 
x\  y'  of  the  triangles  (cT)  and  (e)  are  homologous  to  y^  x 
of  triangle  (a),  we  may  express  the  relations  of  the  last 
article  thus  : 

The  functions  of    -j  ^^o  ,  n  correspond  to  the  same  functions 

of  6^  while  those  of  \  QrT/^o  ,   n  correspond  to  the  co functions 
of  6,  due  attention  being  paid  to  the  signs. 

The  student  can  readily  determine  the  sign  in  any  given 
case,  whether  6  be  acute  or  obtuse,  by  considering  in  what 
quadrant  the  compound  angle,  90°  ±  0,  180°  ±  6,  etc.,  would 


84  PLANE   TRIGONOMETRY. 

lie  if  6  were  an  acute  angle,  and  prefixing  to  the  correspond- 
ing functions  of  0  the  signs  of  the  respective  functions  for 
an  angle  in  that  quadrant.  Thus  90°  +  ^,  if  ^  be  acute,  is 
an  angle  of  the  second  quadrant,  so  that  sine  and  cosecant 
are  plus,  the  other  functions  minus.  It  will  be  seen  that 
sin  (90°  +  (9)=  +  cos  (9,  cos  (90°  +  (9)=  -  sin^,  etc.,  and  this 
will  be  true  whatever  be  the  magnitude  of  6.  It  will  assist 
in  fixing  in  the  memory  these  important  relations  to  notice 
that  when  in  the  compound  angle  6  is  measured  from  the 
?/-axis,  as  in  90°  ±  ^,  270°  ±  ^,  the  functions  of  one  angle 
correspond  to  the  co-functions  of  the  other,  but  when  in  the 
compound  angle  6  is  measured  from  the  a;-axis,  as  in  ±  ^, 
180°  ±  6^  then  the  functions  of  one  angle  correspond  to  the 
same  functions  of  the  other. 

These  relations,  as  has  been  noted  in  Art.  28,  can  be 
extended  to  angles  greater  than  360°,  and  it  may  be  stated 
generally  that 

function  (9  =  ±  function  (2  ^  •  90°  ±  (9), 

function  (9  =  ±  co-function  [(2  n  +  1)  90°  ±  (9]. 

Computation  tables  contain  angles  less  than  90°  only.  The  chief 
utility  of  the  above  relations  will  be  the  reduction  of  functions  of  angles 
greater  than  90°  to  functions  of  acute  angles.  Thus,  to  find  tan  130°  20', 
look  in  the  tables  for  cot  40°  20',  or  for  tan  49°  40'.    Why  ? 

Ex.  1.  What  angles  less  than  360°  have  the  same  numerical  cosine 
as  20°? 

cos  20°  =  -  cos  (180°  ±  20°)  =  cos  (360°  -  20°). 

.-.  200°,  160°,  340°  have  the  same  cosine  numerically  as  20°. 

2.   Find  the  functions  of  135° ;  of  210°. 

sin  135°  =  sin  (90°  +  45°)  =  cos  45°  =  ^v^, 

cos  135°  =  cos  (180°  -  45°)  =  -  cos  45°  =  -  i  V2,  etc. 

sin  210°  =  sin  (180°  +  30°)  =  -  sin  30°  =  -  \. 

Let  the  student  give  the  other  functions  for  each  angle. 


INVERSE  FUNCTIONS.  35 


ORAL  WORK. 

1.  Determine  the  sine  and  tangent  of  each  of  the  following  angles : 
30°,  120°,  -  30°,  -  60°,  f  TT,  2f  tt,  -  135°,  -  ir. 

2.  Which  is  the  greater,  sin  30°  or  sin(-  30°)?  tan  135°  or  tan  45°? 
cos  60°  or  cos(  -  60°)  ?   sin  22°  30'  or  cos  67°  30'  ? 

3.  What  positive  angle  has  the  same  tangent  as  — ?  the  same  sine 
as  50°?  ^ 

4.  If  tan^  =  -l,  findsin^. 

5.  Find  sin  510°,  cos(-  60°),  tan  150°. 

6.  Reduce  in  two  ways  to  functions  of  a  positive  acute  angle,  cos  122° 
tan  140°  30',  sin  (-60°).       ' 

7.  Find  all  positive  values  of  x,  less  than  360°,  satisfying  the  fol- 
lowing equations :  cos  x  —  cos  45°,  sin  2  a:  =  sin  10°,  tan  3  a;  =  tan  60°, 
sin  X  =  sin  30°,  tan  x  =  tan  135°. 

8.  What  angles  are  determined  when  (a)  sine  and  cosine  are  +  ? 
(b)  cotangent  and  sine  are  —  ?  (c)  sine  +  and  cosine  —  ?  (</)  cosine  — 
and  cotangent  +  ? 

INVERSE   FUNCTIONS. 

31.  That  a  is  the  sine  of  an  angle  6  may  be  expressed  in 
two  ways,  viz.,  sin  6  =  a^  or,  inversely,  9  =  sin""i  a,  the  latter 
being  read,  6  equals  an  angle  whose  sine  is  a,  or,  more  briefly, 
0  is  the  anti-sine  of  a. 

The  notation  sin~ia,  cos"^  a,  tan-^a,  etc.,  is  not  a  fortunate  one,  but 
is  so  generally  accepted  that  a  change  is  not  probable.  The  symbol  may 
have  been  suggested  from  the  fact  that  if  ax  =  b,  then  x  =  a~^  b,  whence, 
by  analogy,  if  sin  ^  =  a,  ^  =  sin"i  a.  But  the  likeness  is  an  analogy  only, 
for  there  is  no  similarity  in  meaning.    Sin"^  a  is  an  angle  0,  where  sin  0  =  a, 

and  is  entirely  different  from  (sin  a)-^  = .     In  Europe  the  symbols 

sin  a 
arc  sin  a,  arc  cos  a,  etc.,  are  employed. 

32.  Principal  value.  We  have  found  that  in  sin  6  =  a^ 
for  any  value  of  ^,  a  can  have  but  one  value ;  but  in 
6  =  sin~i  a,  for  any  value  of  a  there  are  an  indefinite  number 
of  values  of  6  (Art.  27,  2). 

Thus,  when  sin  (9  =  a,  if  a  =  J,  (9  may  be  30°,  150°,  390°, 
510°,  -  330°,  etc.,  or,  in  general,  wtt  +(-  1)"30°. 

In  the  solution  of  problems  involving  inverse  functions. 


36 


PLANE   TRIGONOMETRY. 


the  numerically  least  of  these  angles,  called  the  principal 
value^  is  always  used ;  i.e.  we  understand  that  sin~i  a,  tan~i  a, 
are  angles  between  +  90°  and  —  90°,  while  the  limits  of 
cos-la  are  0°  and  180°. 

Thus,     sin-i  i  =  30°,    sin-i(  -  J)  =  -  30°,    cos"!  J  =  60°, 
cos-i(-i)=120°. 

ORAL  WORK. 

How  many  degrees  in  each  of  the  following  angles?    How  many 
radians  ? 


1.  cos-i:^? 

2 

2.  tan-il? 

3.  cot-i(-V3)? 

4.  sin-i(-iV2)? 

5.  cos-i(-iV2)? 

6.  sln-.(_^), 


7.  tan-iVS? 

8.  cos-iQ? 

9.  sin-n? 

10.  tan-iQ? 

11.  tan-i(-l)? 

12.  sin-i(-l)? 


Find  the  values  of  the  functions : 

13.  sin(tan-ii\/3). 

14.  tan(cos"^  1). 

15.  tan(cot~i[—  go]). 

16.  cos(tan~iGo). 

17.  sin(sin-i|V2). 

18.  tan(tan-ia;). 


19.  cos(sin-iO). 

20.  sin(cos-i[-  1]). 

21.  cos(cot-iV3). 

22.  tan(sin-i[-l]). 

23.  sin(tan-i[-l]). 

Ex.  1.   Construct  cot-^  f . 
Construct  the  right  triangle  xyr,  so  that  a:  =  4, 
2/  =  3,  whence  angle  xr  =  cot"^  f . 

2.   Find  cos(tan-i  ^^). 
Let  6  =  tan-i  j^,  whence 

tan  0  =  xV>  and  cos  0  =  |f 
.♦.  cos  6  =  cos(tan-^  -j^)  =  if. 


and 


3.  If  ^  =  csc-i  a,  prove  6  =  cos~i  — ^ 
CSC  ^  =  a 
cos  e  =Vl--„  =  ^"'^  ~  \  or  ^  =  cos- 


.  sin  ^  =  -> 
a 


.1  Va^ 


EXAMPLES.  37 


EXAMPLES. 

1.  Construct  sin-^f,  tan-ij^,  cos-i(—  ^). 

2.  Find  tan(sin~ix^j),  sin(tan-iy\). 

3.  If  ^  =  sin-i  a,  prove  0  =  tan-^ — 


VI -a2 

4.  Show  that  sin"^  a  =  90°  —  cos"^  a. 

5.  Prove  tan-i\/3  +  cot-iV3=^. 

6.  Prove  tan-ifsin  '^\  =  cos-^^^. 

7.  What  angles,  less  than  360°,  have  the  same  tangent  numerically 
as  10°? 

8.  Given  tan  143°  22'  =  -  0.74357 ;  find,  correct  to  0.00001,  sine  and 
cosine. 

9.  If  cot2(90°  +  /?)  +  csc(90°  -  /8)  -  1  =  0,  find  tan  fi. 

10.  Find  all  positive  values  of  x,  less  than  360°,  when  sin  x = sin  22°  30' ; 
when  tan  2  a;  =  tan  60°. 

11.  When  is  sin  x  = possible,  and  when  impossible  ? 

12.  Verify  sin-i  |  +  cos'i—  +  tan-i  V3  =  sin-i  ^. 

13.  What  values  of  x  will  satisfy  sin-i(.r2  -  x)=  30°  ? 

14.  If  tan2  e  -  sec2  a  =  1,  prove  sec  $  +  tan^  ^  esc  ^  =  (3  +  tan2  a)^. 

15.  Prove  sin  ^  (1  +  tan  A)+  cos  ^ (1  +  cot  A)  =  sec  ^  +  esc  A. 

16.  Solve  the  simultaneous  equations : 

sin-i(2  X  +  Sy)=30°  and  3  a:  +  2 y  =  2. 


17.  Verify  (a)    tan60°  =  V^  -cosli 

'  1   +  cos  1'.: 

(6)    cos  60°  = 


i2!. 
120°' 

1  -tan2  30° 
l+tan2  30°* 

(c)    2  sin2  60°  =  1  -  cos  120°. 


18.   Show  that  the  cosine  of  the  complement  of  -  equals  the  sine  of 

6 
the  supplement  of  -• 


38  PLANE   TRIGONOMETRY. 

REVIEW. 

Before  leaving  a  problem  the  student  should  review  and  master  all 
principles  involved. 

1.  Construct  cos'^xV 5  sin-i(— |);  tan-i2. 

2.  Find  cos  (sin-i  f ) ;   tan  (cos-i  [  —  i]  ) . 

3.  Prove  cot"^  a  =  cos~^^ 

VI  +a2 

4.  Given  a  =  cot-i|,  find  tan  a  +  sin  (90°  +  a). 

5.  Find  tan  (  sin-i|  +  cos-^: — )• 

6.  State  the  fundamental  relations  between  the  trigonometric  func- 
tions in  terms  of  the  inverse  functions.     Thus, 

1 


sin~i«  =  csc~^-,    sin~ia  =  cos~^Vl  —  aK  etc. 
a 

7.  Find  all  the  angles,  less  than  360°,  whose  cosine  equals  sin  120°. 

8.  Given  cot~i  2.8449,  find  the  sine  and  cosine  of  the  angle,  correct 
to  0.0001. 

9.  If  tan2  (180°  -0)-  sec  (180°  +  (9)  =  5,  find  cos  0. 

irx    T£    •    n      9  £    J  tan^^  +  cos^^ 

10.  If  sm  6  =  ^,  find  -— ' -- 

^'  tan2^-cos2^ 

11.  Is  sin  X  —  2  cos  x  +  Ssina;  —  6  =  0a  possible  equation  ? 

12.  Verify  (a)     sin  60°=     ^  tan  30°    , 

^   ^  ^  l  +  tan230° 

(b)  2  cos2  60°  =  1  +  cos  120°. 

(c)  cos  60°  -  cos  90°  =  2  cos2  30°  -  2  cos2  45°. 

13.  If  sin  X  =  — ^K^_± 1 —  find  sec  x  and  tan  x. 

a^  +  2ab  +  2  b^ 

14.  Prove   1  +  sin  ^  -  cos  ^  _^  1  +  sin^  +  cos^^  ^^^^^^ 

1  +  sin  ^  +  cos  $     1  +  sin  ^  —  cos  0 

15.  Prove 

cos  45°  +  cos  135°  +  cos  30°  +  cos  150°  -  cos  210°  +  cos  270°  =  sin  60°. 

16.  If  tan  0  =  prove  that 

Va2  _  Ij2 

sin  ^(1  +  tan  6)  +  cos  ^(1  +  cot  ^  -  sec 0  =  |- 

17.  Solve  sin2  x  +  sin^  (x  +  90°)  +  sin2  (^  ^  i80°)  =  1. 


EXAMPLES.  39 

18.  Given  cos^  a  =  msina  —  n,  find  sin  a. 

19.  If  sin2y3=-A^,  find^. 

2  sec  p 

20.  Given  tan  238°  =1.6,  find  sin  148°. 

21.  Prove  tan-i  m  +  cot-i  m  =  90°. 

22.  Find  sin  (sin~ij9  +  cos~ijo). 

23.  Solve  cot2  ^  (2  esc  ^  -  3)  +  3  (esc  ^  -  1)  =  0. 

24.  Prove  sin^  a  sec^  ^  +  tan^  /?  cos^  a  =  sin^  a  +  tan2  p. 

25.  Prove  cos^  F  +  sin^  F  =  1  -  3  sin^  F  +  3  sin^  F. 

26.  What  values  of  A  satisfy  sin  2  A  =  cos  3  ^  ? 


27.  If  tan  C  =  ^^  ~  '^'^,  and  tan  D  =\^  -  cos  C^  ^^^^  ^^^  ^  .^  ^^^^^^ 
ofm.  ''^  M+cosC 

28.  If  sin  a:  —  cos  x  +  4  cos^  a:  =  2,  find  tan  x ;  sec  a:. 

29.  Does  the  value  of  sec  x,  derived  from  sec^  x  =  — — - — - — ,  give  a 
possible  value  of  a:?  1  -  cos  x 

30.  Prove 

[cot  (90°  -  ^  )  -  tan  (90°  +  A)]  [sin  (180°  -A)  sin  (90°  +  /I )]  =  1 . 

31.  Prove  (1  +sin^)2[cot^  +2sec^(l  -csc^)]  +  csc^  cos^^  =  0. 

32.  Given  sin  a:  =  m  sin  y,  and  tan  x  =  n  tan  y,  find  cos  x  and  cos  y. 

33.  Given  cot  201°  =  2.6,  find  cos  111°. 

34.  Find  the  value  of 

cos-H  +  sin-HV^  +  csc-i(-  1)+  tan-U  -  2cot-iV3. 

35.  Solve  2  cos^d  +  11  sin  ^  -  7  =  0. 

36.  Prove 

cos2  B  +  cos2  {B  +  90°)  +  cos2  (5  +  180°)  +  cos2(5  +  270°)  =  2. 


CHAPTER   IV. 

COMPUTATION   TABLES. 

33.  Natural  functions.  It  has  been  noted  that  the  trigo- 
nometric functions  of  angles  are  numbers^  but  the  values 
were  found  for  only  a  few  angles,  viz.  0°,  30°,  45°,  60°, 
90°,  etc.  In  computations,  however,  it  is  necessary  to  know 
the  values  of  the  functions  of  any  angle,  and  tables  have 
been  prepared  giving  the  numerical  values  of  the  functions 
of  all  angles  between  0°  and  90°  to  every  minute.  In 
these  tables  the  functions  of  any  given  angle,  and  co^i- 
versely  the  angle  corresponding  to  any  given  function,  can 
be  found  to  any  required  degree  of  accuracy ;  e.g.  by  look- 
ing in  the  tables  we  find  sin  24°  26'=  0.41363,  and  also 
1 .6415  =  tan  58°  39'.  These  numbers  are  called  the  natural 
functions.,  as  distinguished  from  their  logarithms,  which  are 
called  the  logarithmic  functions  of  the  angles. 

Ex.  1.   Find  from  the  tables  of  natural  functions : 

sin35n4';     cos  54°  46';     tan  78°  29';     cos  112°  58';     sin  135°. 

2.   Find  the  angles  less  than  180°  corresponding  to : 
sin-i  0.37865;  cos-i  0.37865;   tan-i  0.58670 ;   cos"!  0.00291 ;   sin-^O 


34.  Logarithms.  The  arithmetical  processes  of  multi- 
plication, division,  involution,  and  evolution,  are  greatly 
abridged  by  the  use  of  tables  of  logarithms  of  numbers 
and  of  the  trigonometric  ratios,  which  are  numbers.  The 
principles  involved  are  illustrated  in  the  following  table : 

Write  in  parallel  columns  a  geometrical  progression  having 
the  ratio  2,  and  an  arithmetical  progression  having  the  dif- 
ference 1,  as  follows  : 

40 


LOGARITHMS. 


41 


G.  P. 

A.  P. 

1 

0 

2 

1 

4 

2 

8 

3 

16 

4 

32 

5 

64 

6 

128 

7 

256 

8 

512 

9 

1024 

10 

2048 

11 

4096 

12 

8192 

13 

16384 

14 

32768 

15 

655S6 

16 

131072 

17 

262144 

18 

524288 

19 

1048576 

20 

It  will  be  perceived  that  the  numbers  in 
the  second  column  are  the  indices  of  the 
powers  of  2  producing  the  corresponding 
numbers  in  the  first  column,  thus  :  2^  =  64, 
211  =  2048,  218  =  262144,  etc.  The  use  of 
such  a  table  will  be  illustrated  by  examples. 

Ex.  1.    Multiply  8192  by  128. 

From  the  table,  8192  =  2^%  128  =  2'.  Then  by 
actual  multiplication,  8192  x  128  =  1048576,  or  by  the 
law  of  indices,  21^  x  2^  =  220  =  1048576  (from  table). 

Notice  that  the  simple  operation  of  addition  is  sub- 
stituted for  multiplication  by  adding  the  numbers  in 
the  second  column  opposite  the  given  factors  in  the 
first  column.  This  sum  corresponds  to  the  number 
in  the  first  column  which  is  the  required  product. 

2.   Divide  16384  by  512. 

16384  -4-  512  =  32,  which  corresponds  to  the  result 
obtained  by  use  of  the  table,  or  2^^  -  2^  =  2^  =  32. 
The  operation  of  subtraction  takes  the  place  of 
division. 


3.    Find  V262144. 


>>^62144 


2^^  =  03  — 


In  the  table,  262144  is  opposite  18.  18  --  6  =  3, 
which  is  opposite  8,  the  required  root ;  i.e.  simple  division  takes  the 
place  of  the  tedious  process  of  evolution. 

4.    Cube  64.  6.    Find  ^^^2768. 


5.   Multiply  256  by  4096. 


7.   Divide  1048576  by  32768. 


35.  The  above  table  can  be  made  as  complete  as  desired 
by  continually  inserting  between  successive  numbers  in  the 
first  column  the  geometrical  mean,  and  between  the  opposite 
numbers  in  the  second,  the  arithmetical  mean,  but  in  prac- 
tice logarithms  are  computed  by  other  methods.  The  num- 
bers in  the  second  column  are  called  the  logarithms  of  the 
numbers  opposite  in  the  first  column.  2  is  called  the  base  of 
this  system,  so  that  the  logarithm  of  a  number  is  the  exponent 
by  which  the  base  is  affected  to  produce  the  number. 


42  PLANE   TRIGONOMETRY. 

Thus,  the  logarithm  of  512  to  the  base  2  is  9,  since 
29  =  512. 

Logarithms  were  invented  by  a  Scotchman,  John  Napier,  early  in  the 
seventeenth  century,  but  his  method  of  constructing  tables  was  different 
from  the  above.  See  Encyc.  Brit,  art.  ^'•Logarithms,''''  for  an  exceedingly 
interesting  account.  De  Morgan  says  that  by  the  aid  of  logarithms  the 
labor  of  computing  has  been  reduced  for  the  mathematician  to  about 
one-tenth  part  of  the  previous  expense  of  time  and  labor,  while  Laplace 
has  said  that  John  Napier,  by  the  invention  of  logarithms,  lengthened 
the  life  of  the  astronomer  by  one-half. 

Columns  similar  to  those  above  might  be  formed  with  any 
other  number  as  base.  For  practical  purposes,  however,  10 
is  always  taken  as  the  base  of  the  system,  called  the  common 
system^  in  distinction  from  the  natural  system^  of  which  the 
base  is  2.71828  •••,  the  value  of  the  exponential  series  (^Higher 
Algebra) .  The  natural  system  is  used  in  theoretical  discus- 
sions. It  follows  that  common  logarithms  are  indices^  positive 
or  negative^  of  the  powers  of  10. 

Thus,  103  =  1000  ;  i.e.  log  1000  =  3  ; 

10-2  =  i-  =  0.01;  i.e.  log0.01  =  -2. 

36.  Characteristic  and  mantissa.  Clearly  most  numbers 
are  not  integral  powers  of  10.  Thus  300  is  more  than  the 
second  and  less  than  the  third  power  of  10,  so  that 

log  300  =  2  plus  a  decimal. 

Evidently  the  logarithms  of  numbers  generally  consist  of 
an  integral  and  a  decimal  part,  called  respectively  the  charac- 
teristic and  the  mantissa  of  the  logarithms. 

37.  Characteristic  law.  The  characteristic  of  the  loga- 
rithm of  a  number  \^  independent  of  the  digits  composing 
the  number,  but  depends  on  the  position  of  the  decimal 
point,  and  is  found  by  counting  the  number  of  places  the  first 
significant  figure  in  the  number  is  removed  from  the  units'' 
place,  being  positive  or  negative  according  as  the  first  significant 


LOGARITHMS.  43 

figure  is  at  the  left  or  the  right  of  units'  place.  This  follows 
from  the  fact  that  common  logarithms  are  indices  of  powers 
of  10,  and  that  10",  n  being  a  positive  integer,  contains  n  -f-  1 
places,  while  10~"  contains  n—1  zeros  at  the  right  of  units' 
place.  Thus  in  146.043  the  first  significant  figure  is  two 
places  at  the  left  of  units'  place  ;  the  characteristic  of  log 
146.043  is  therefore  2.  In  0.00379  the  first  significant  digit 
is  three  places  at  the  right  of  units'  place,  and  the  charac- 
teristic of  log  0.00379  is  -  3. 

To  avoid  the  use  of  negative  characteristics,  such  charac- 
teristics are  increased  by  10,  and  —10  is  written  after  the 
logarithm.  Thus,  instead  of  log  0.00811  =  3.90902,  write 
7.90902  —  10.  In  practice  the  —  10  is  generally  not  written, 
but  it  must  ahvays  be  remembered  and  accounted  for  in  the 
result. 

Ex.   Determine  the  characteristic  of  the  logarithm  of : 
1;  46;  0.009;  14796.4;  230.001;  lO^  x  76;  0.525;  1.03;  0.000426. 

38.  Mantissa  law.  The  mantissa  of  the  logarithm  of  a 
number  is  hidependent  of  the  position  of  the  decimal  point, 
but  depends  on  the  digits  composing  the  number,  is  always 
positive^  and  is  found  in  the  tables. 

For,  moving  the  decimal  point  multiplies  or  divides  a 
number  by  an  integral  power  of  10,  i.e.  adds  to  or  subtracts 
from  the  logarithm  an  integer,  and  hence  does  not  affect  the 
mantissa.     Thus, 

log   225.67  =  log  225.67, 

log   2256.7  =  log  225.67  X  101    =  log  225.67 -f  1, 
log 22567.0  =  log  225.67  x  102    ^  i^g  225.67  +  2, 
log   22.567  =  log 225.67  x  lO-i  =  log 225.67 +(- 1), 
log  0. 22567  =  log  225. 67  x  10-3  =  log  225. 67  +  (  -  3), 

so  that  the  mantissae  of  the  logarithms  of  all  numbers  com- 
posed of  the  digits  22567  in  that  order  are  the  same,  .35347. 
MoviTig  the  decimal  point  affects  the  characteristic  only. 
The  student  must  remember  that  the  mantissa  is  always  positive. 


44  PLANE  TRIGONOMETRY. 

Log  0.0022567  is  never  written  -  3  +.35347,  but  3.35347,  the  minus 
sign  being  written  above  to  indicate  that  the  characteristic  alone  is  nega- 
tive. In  computations  negative  characteristics  are  avoided  by  adding 
and  subtracting  10,  as  has  been  explained. 

39.  We  may  now  define  the  logarithm  of  a  number  as  the 
index  of  the  power  to  which  a  fixed  number,  called  the  base, 
must  be  raised  to  produce  the  given  number. 

Thus,  a^  =  5,  and  x  =  logab  (where  log  J)  is  read  logarithm 
of  b  to  the  base  a')  are  equivalent  expressions.  The  relation 
between  base,  logarithm,  and  number  is  always 

(base)^°^  =  number. 

To  illustrate:  log28  =  3  is  the  same  as  2^  =  8;  log381  =  4  and 
3*=  81  are  equivalent  expressions  ;  and  so  are  log^QlOOO  =  3 
and  103  =  1000,  and  logio0.001=  -3  and  10-3=  0.001. 

Find  the  value  of  : 
log464;  log5l25;  log3243;  log«(«)^;  log27  3  ;  log^l. 

40.  From  the  definition  it  follows  that  the  laws  of  indices 
apply  to  logarithms,  and  we  have : 

I.  The  logarithm  of  a  product  equals  the  sum  of  the  loga- 
rithms of  the  factors. 

II.  The  logarithm  of  a  quotient  equals  the  logarithm  of  the 
dividend  minus  the  logarithm  of  the  divisor. 

III.  The  logarithm  of  a  power  equals  the  index  of  the 
power  times  the  logarithm  of  the  number. 

IV.  The  logarithm  of  a  root  equals  the  logarithm  of  the 
number  divided  by  the  index  of  the  root.  , 

For  if  a^  =^n  and  a^  =  m, 

■  then  n  xm  =  a^"*"^    .*.  log nm  =  x-\-y  =  log n  +  log w; 

and  n-^m  =  a^~y,   .-.log—   =  a;  — ^  =  logn  — logm; 

m 

also  n""—  (a^y=  a^^     .-.  log n""   =rx  =  r  xlogn; 

_  Z_  _  -J 

finally,    Vn  =  Va^  =  a%      .*.  log</n  =  -  =  -  log  w. 


LOGARITHMS.  45 


EXAMPLES. 
Given  log  2  =  0.30103,  log  3  =  0.47712,  log  5  =  0.69897,  find : 

10.   logV||. 
11, 


1.   log  4. 

4.   log  9. 

7.   log  153. 

2.   log  6. 

5.   log  25. 

8.   logf. 

3.   log  10. 

6.   logVS. 

9.   logl5x9. 

•-Vl 


X  o** 


xlO 


USE  OF  TABLES. 
41.  To  find  the  logarithm  of  a  number. 

First.    Find  the  characteristic,  as  in  Art.  37. 

Second.    Find  the  mantissa  in  the  tables,  thus : 

(a)  When  the  number  consists  of  not  more  than  four 
figures. 

In  the  column  N  of  the  tables  find  the  first  three  figures, 
and  in  the  row  N  the  fourth  figure  of  the  number.  The 
mantissa  of  the  logarithm  will  be  found  in  the  row  opposite 
the  first  three  figures  and  in  the  column  of  the  fourth  figure. 

Illustration.    Find  log  42.38. 

The  characteristic  is  1.     (Why  ?) 

In  the  table  in  column  N  find  the  figures  423,  and  on  the 
same  page  in  row  N  the  figure  8.  The  last  three  figures  of 
the  mantissa,  716,  lie  at  the  intersection  of  column  8  and 
row  423.  To  make  the  tables  more  compact  the  first  two 
figures  of  the  mantissa,  62,  are  printed  in  column  0  only. 
Then  log42.38  =  1.62716. 

Find  log  0.8734  =  1.94121, 

log  3.5        =  log  3.500  =  0.54407, 
log  36350  =4.56050. 

(5)  When  the  number  consists  of  more  than  four  figures. 

Find  the  mantissa  of  the  logarithm  of  the  number  com- 
posed of  the  first  four  figures  as  above.  To  correct  for  the 
remaining  figures  we  interpolate  by  means  of  the  principle  of 
proportional  parts,  according  to  which  it  is  assumed  that,  for 
differences  small  as  compared  with  the  numbers,  the  diff'ereiices 


46  PLANE  TRIGONOMETRY. 

hetiveen  several  numbers  are  proportional  to  the  differences  be^ 
tween  their  logarithms. 

The  theorem  is  only  approximately  correct,  but  its  use 
leads  to  results  accurate  enough  for  ordinary  computations. 

Ex.  1.    To  find  log  89.4562. 

As  above,  mantissa  of  log  894500  =  0.95158, 

mantissa  of  log  894600  =  0.95163, 

.-.  log  894600  -  log  894500  =  O.OOOOo,  called  the  tabular  difference. 

Let  log  894562  -  log  894500  =  x  hundred-thousandths. 

Now,  by  the  principle  of  proportional  parts, 

log  894562  -  log  894500  ^  894562  -  894500 
log  894600  -  log  894500  ~  894600  -  894500' 

X       6'^ 
or         -  =  — ^,  whence  x  =  .62  of  5  =  3.1 
5      100 

.-.  log  89.4562  =  1.95158  +  0.00003  =  1.95161, 

all  figures  after  the  fifth  place  being  rejected  in  five-place  tables.  If, 
however,  the  sixth  place  be  5  or  more,  it  is  the  practice  to  add  1  to  the 
figure  in  the  fifth  place.  Thus,  if  a;  =  0.0000456,  we  should  call  it 
0.00005,  and  add  5  to  the  mantissa. 

2.  Find  log  537.0643. 

To  interpolate  we  have  x  :  9  =  643  :  1000,  i.e.  x  =  5.787 ; 
.-.  log  537.0643  =  2.72997  -f  0.00006. 

3.  Find  log  0.0168342  =  2.22619. 

4.  Find  log  39642.7  =  4.59816. 

42.   To  find  the  number  corresponding  to  a  given  logarithm. 

The  characteristic  of  the  logarithm  determines  the  posi- 
tion of  the  decimal  point  (Art.  37). 

(a)  If  the  mantissa  is  in  the  tables,  the  required  number 
is  found  at  once. 

Ex.  1.  Find  log~^  1.94621  (read,  the  number  whose  logarithm  is 
1.94621). 

The  mantissa  is  found  in  the  tables  at  the  intersection  of  row  883  and 
column  5. 

.-.  log-i  1.94621  =  88.35, 

the  characteristic  1  showing  that  there  are  two  integral  places. 


LOGARITHMS.  47 

(5)  If  the  exact  mantissa  of  the  given  logarithm  is  not  in 
the  tables,  the  first  four  figures  of  the  corresponding  num- 
ber are  found,  and  to  these  are  annexed  figures  found  by 
interpolating  by  means  of  the  principle  of  proportional 
parts,  as  follows : 

Find  the  two  successive  mantissas  between  which  the  given 
mantissa  lies.  Then,  by  the  principle  of  proportional  parts, 
the  amount  to  be  added  to  the  four  figures  already  found  is 
such  a  part  of  1  as  the  difference  between  the  successive 
mantissse  is  of  the  difference  between  the  smaller  of  them 
and  the  given  mantissa. 

2.  Find  log-i  1.43764. 

Mantissa  of  log  2740  =  0.43775 

of  log  2739  =  0.43759 
Differences  1  16 

Mantissa  of  log  required  number  =  0.43764 

of  log2739  =  0.43759 

Differences  x  5 

By  p.  p.  a;  :  1  =  5  :  16  and  x  =  ^^-  0.3125. 

Annexing  these  figures,  log-i  1.43764  =  27.3931+. 

3.  Find  log-i  T.48762. 

The  differences  in  logarithms  are  14,  6. 

...  a:  =  A  =  .4284-, 

14: 

and  log-i  1.48762  =  0.307343+. 

4.  Find  log  891.59;  log  0.023;  log^;  log  0.1867;  log  V2. 

5.  Find  log-i  2.21042  ;  log-i  0.55115;  log-i  1.89003. 

43.  Logarithms  of  trigonometric  functions.  These  might 
be  found  by  first  taking  from  the  tables  the  natural  func- 
tions of  the  given  angle,  and  then  the  logarithms  of  these 
numbers.  It  is  more  expeditious,  however,  to  use  tables 
showing  directly  the  logarithms  of  the  functions  of  angles 
less  than  90°  to  every  minute.  Functions  of  angles  greater 
than  90°  are  reduced  'to  functions  of  angles  less  than  90°  by 


48  PLANE   TRIGONOMETRY. 

the  formulae  of  Art.  29.  To  make  the  work  correct  for 
seconds,  or  any  fractional  part  of  a  minute,  interpolation 
is  necessary  by  the  principle  of  proportional  parts,  thus  : 

Ex.  1.   Find  log  sin  28°  32'  21". 

In  the  table  of  logarithms  of  trigonometric  functions,  find  28°  at  the 
top  of  the  page,  and  in  the  minute  column  at  the  left  find  32'.  Then 
under  log  sin  column  find  log  sin  28°  32'  =  9.67913  -  10 

log  sin  28°  33  =  9.67936  -  10 
Differences         1'  23 

By  p.  p.  a; :  23  =  21"  :  60",  i.e.  a;  =  —  x  23  =  8.4. 

60 

.-.  log  sin  28°  32'  21"  =  9.67913  +  0.00008  -  10 

=  9.67921  -  10. 

Whenever  functions  of  angles  are  less  than  unity,  i.e.  are  decimals 
(as  sine  and  cosine  always  are,  except  when  equal  to  unity,  and  as  tan- 
gent is  for  angles  less  than  45°),  the  characteristic  of  the  logarithm  will 
be  negative,  and,  accordingly,  10  is  always  added  in  the  tables,  and  it 
must  be  remembered  that  10  is  to  be  subtracted.  Thus,  in  the  example 
above,  the  characteristic  of  the  logarithm  is  not  9,  but  1,  and  the  log- 
arithm is  not  9.67913,  as  written  in  the  tables,  but  9.67913  -  10. 

2.   Find  log  cos  67°  27' 50". 

In  the  table  of  logarithms  at  the  foot  of  the  page,  find  67°,  and  in  the 
minute  column  at  the  right,  27'.  Then  computing  the  difference  as 
above,  x  =  25. 

But  it  must  be  noted  that  cosine  decreases  as  the  angle  increases 
toward  90°.  Hence,  log  cos  67°  27' 50"  is  less  than  log  cos  67°  27',  i.e. 
the  difference  25  must  be  subtracted,  so  that 

log  cos  67°  27'  50"  =  9.58375  -  0.00025  -  10 
=  9.58350  -  10. 

44.  To  find  the  angle  when  the  logarithm  is  given,  find  the 
successive  logarithms  between  which  the  given  logarithm 
lies,  compute  by  the  principle  of  proportional  parts  the 
seconds,  and  add  them  to  the  less  of  the  two  angles  corre- 
sponding to  the  successive  logarithms.  This  will  not  neces- 
sarily be  the  angle  corresponding  to  the  less  of  the  two 
logarithms  ;  for,  as  has  been  seen,  the  number,  and,  therefore, 
the  logarithm,  may  decrease  as  the  angle  increases. 


LOGARITHMS.  49 

Ex.  1.   Find  the  angle  whose  log  tan  is  9.88091. 

log  tan  37°  14'  =  9.88079  -  10 
log  tan  37°  15'  =  9.88105  -  10 

Differences        60"  26 

log  tan  37°  14'  =  9.88079  -  10 
log  tan  angle  required  =  9.88091  —  10 

Differences  x"  12 

.-.  a: :  60  =  12  :  26,    or    x"  =  if  x  60"  =  28",   approximately,   and  the 
angle  is  37°  14'  28". 

2.  Find  the  angle  whose  log  cos  =  9.82348. 

We  find  x  =  ^x  60"  =  26",  and  the  angle  is  48°  14'  26". 

3.  Show  that        log  cos    25°  31' 20"  =  9.95541 ; 

log  sin  110°  25'  20"  =  9.97181 ; 
log  tan  49°  52'  10"  =  0.07418. 

4.  Show  that  the  angle  whose  log  tan  is  9.92501  is  40°  4'  40" ;  whose 
log  sin  is  9.88365  is  49°  54'  20" ;  whose  log  cos  is  9.50828  is  71°  11'  50". 

45.   Cologarithms.     In  examples  involving  multiplications 
and  divisions  it  is  more  convenient,  if  n  is  any  divisor,  to 

add  log  -  than  to  subtract  log  n.     The  logarithm  of  -  is 
called  the  cologarithm  of  n.     Since 

log  -  =  log  1  —  log  n=0  —  log n, 

it  follows  that  colog n  =  —  log  n^  i.e.  logn  subtracted  from 
zero.     To  avoid  negative  results,  add  and  subtract  10. 


Ex.1.  Find  colog  2963. 

log  1  =  10.00000  - 
log  2963=    3.47173 

-10 

.-.  colog  2963=    6.52827- 

-10 

2.  Find  colog  tan  16°  17'. 

log  1  =  10.00000  - 
log  tan  16°  17'=    9.46554- 

-10 
-10 

.-.  colog  tan  16°  17'  =    0.53446 


50  PLANE   TRIGONOMETRY. 

By  means  of  the  definitions  of  the  trigonometric  functions,  the  parts 
of  a  right  triangle  may  be  computed  if  any  two  parts,  one  of  them  being 
a  side,  are  given.     Thus, 

■  B    given  a  and  A  in  the  rt.  triangle  ABC. 

Then      c  =  a  -^  sin  A,  b  =  a  ^  tan  A , 

B  =  90°  -A. 

Again,  if  a  and  b  are  given,  then 

tan  ^  =-,c  =  a^  sin  A^  and  B  =  90°-A- 
b 

3.  Given  c  =  25.643,  B  =  37°  25'  20",  compute  the  other  parts. 

^  =  90°  -  37°  25'  20"  =  52°  34'  40". 

a  =  c  cos  B.  b  =  a  tan  B. 

log  c  =  1.40897  log  a  =  1.30889 

log  cos  B  =  9.89992  log  tan  B  =  9.88376 

log  a  =  1.30889  log  b  =  1.19265 

.-.  a  =  20.365.  .-.  b  =  15.583. 

Check:  c^  =  a^ -\- b^  =  20.365^  +  15.583^  =  657.57  =  25.6432. 

4.  Given  b  =  0.356,  B  =  63°  28'  40",  compute  the  other  parts. 

A  =  26°  31'  20". 

h  h 

a  = 


sin  B  tan  B 

log  b  =  9.55145  log  b  =  9.55145 

colog  sin  B  =  0.04829  colog  tan  B  =  9.6981B 

log  c  =  9.59974  log  a  =  9.24961 

c  =  0.3979  a  =  0.1777 

Check:  c^  -  a^  =  0.1583  -  0.03157  =  0.12673  =  b^ 

EXAMPLES. 

Compute  the  other  parts : 

1.  Given   a  =  9.325,  A  =  43°  22'  35". 

2.  Given    c  =  240.32,  a  =  174.6. 

3.  Given  5  =  76°  14' 23",  a  =  147.53. 

4.  Given  a  =  2789.42,  b  =  4632.19. 

5.  Given  c  =  0.0213,  A  =  23°  14". 

6.  Given  &  =  2,  c  =  3. 


CHAPTER  V. 


APPLICATIONS. 


46.  Many  problems  in  measurements  of  heights  and  dis- 
tances may  be  solved  by  applying  the  preceding  principles. 
By  means  of  instruments  certain  distances  and  angles  may 
be  measured,  and  from  the  data  thus  determined  other 
distances  and  angles  computed.  The  most  common  instru- 
ments are  the  chain^  the  transit^  and  the  compass. 

The  chain  is  used  to  measure  distances.  Two  kinds  are  in 
use,  the  engineer'' s  chain  and  the  Gunter^s  chain.  They  each 
contain  100  links,  each  link  in  the  engineer's  chain  being 
12  inches  long,  and  in  the  Gunter's  7.92  inches. 


Fig.  26. 


The  transit  is  the  instrument  most  used  to  measure  hori- 
zontal angles,  and  with  certain  attachments  to  measure  verti- 
cal angles.     The  figure  shows  the  form  of  the  instrument. 


51 


52 


PLANE   TRIGONOMETRY. 


The  mariner^  8  compass  is  used  to  determine  the  directions, 
or  hearings,  of  objects  at  sea.  Each  quadrant  is  divided 
into  8  parts,  making  the  32  points  of  the  compass,  so  that 
each  point  contains  11°  15^ 


Z^A 


Fig.  27. 


tS^ 


Fig.  28. 


47.  The  angle  between  the  horizontal  plane  and  the  line 
of  vision  from  the  eye  to  the  object  is  called  the  angle  of 

elevation,  or  of  depression,  according 
as  the  object  is  above  or  below  the 
Eievaiiony^^^    observer. 

It  is  evident  that  the  elevation 
angle  of  B,  as  seen  from  A,  is  equal 
to  the  depression  angle  of  J.,  as  seen  from  B,  so  that  in  the 
solution  of  examples  the  two  angles  are  interchangeable. 

PROBLEMS. 

48.  Some   of   the   more   common    problems   met  with   in 
practice  are  illustrated  by  the  following : 

To  find   the    height    of  an   object 
when  the  foot  is  accessible. 

The  distance  BC,  and  the  eleva- 
tion angle  B  are  measured,  and  x 
is    determined    from    the    relation  ^ 
X  =  BC  tan  B,  Fig.  29. 


APPLICATIONS. 


63 


Ex.  1.   The  elevation  angle  of  a  cliff  measured  from  a  point  300  ft. 
from  its  base  is  found  to  be  30°.     How  high  is  the  cliff? 


Then 


BC  =  300,  B  =  30°. 

a;  =  300  •  tan  30°  =  300  •  ^  V3  =  100  V3. 


2.  From  a  point  175  ft.  from  the  foot  of  a  tree  the  elevation  of  the 
top  is  found  to  be  27°  19'.     Find  the  height  of  the  tree. 

The  problem  may  be  solved  by  the  use  of  natural  functions,  or  of 
logarithms.  The  work  should  be  arranged  for  the  solution  before  the 
tables  are  opened.     Let  the  student  complete. 


Then 


BC  = 

175. 

B-- 

=  27°  19'. 

x  =  BC  tan  B. 

Or  by 

natural  functions, 

logBC  = 

BC  =  175 

log  tan  B  = 

tan  £  =  0.5165 

logx  = 

.-.  X  =  90.3875. 

.-.  X  =  90.39. 

To  find  the  height  of  an  object 
when  the  foot  is  inaccessible. 

Measure  BB\  6  and  0'. 


Then     x  = 


BC      BB'-hB'C 


cot  6  cot  9 

But  B'  C  =  x  cot  6',  whence  substituting, 

BB' 

cot  6-  cot  6'" 

which  is  best  solved  by  the  use  of  the  natural  functions  of 
e  and  6'. 


3.  Measured  from  a  certain  point  at  its  base  the  elevation  of  the 
peak  of  a  mountain  is  60°.  At  a  distance  of  one  mile  directly  from  this 
point  the  elevation  is  30°.     Find  the  height  of  the  mountain. 

BB'  =  5280  ft.,    e  =  30°,    0'  =  60°. 


^  ^  y  +  5280 
cot  30° 
5280 


But  y  =  xcot 60°. 


X  = 


cot  30°  -  cot  60' 


=  4572.48  ft. 


54 


PLANE   TRIGONOMETRY. 


In  surveying  it  is  often  necessary  to  make  measurements 
across  a  stream  or  other  obstacle  too  wide  to  be  spanned  by 
a  single  chain. 

To  find  the  distance  from  O  to  a 
point  B  on  the  opposite  side  of  a 
stream. 

At  O  measure  a  right  angle,  and 
take     CA    a     convenient     distance. 
Measure  angle  A^  then 
Fi«-3i.       "  BC=CA.t^nA. 

4.  Find  CB  when  angle  A  =  47°  16',  and  CA  =  250  ft. 

5.  From  a  point  due  south  of  a  kite  its  elevation  is  found  to  be 
42°  30';  from  a  point  20  yds.  due  west  £> 

from  this  point  the  elevation  is  36°  24'. 
How  high  is  the  kite  above  the  ground  ? 

^^  =  a:,  cot  42°  30', 

^C  =  re.  cot  36°  24', 

AC^-AB^  =  BC^  =  400. 
.'.  a;2  (cot2  36°  24'  -  cot^  42°  30')  =  400, 
whence 
^2  _  J00_   and  a:  =  .f^  =  24.84  yds. 


.6489 


.805 


Fig.  32. 


EXAMPLES. 

1..  What  is  the  altitude  of  the  sun  when  a  tree  71.5  ft.  high  casts 
a  shadow  37.75  ft.  long  ? 

2.  What  is  the  height  of  a  balloon  directly  over  Ann  Arbor  w^hen 
its  elevation  at  Ypsilanti,  8  miles  away,  is  10°  15'? 

3.  The  Washington  monument  is  555  ft.  high.  How  far  apart  are 
two  observers  who,  from  points  due  east,  see  the  top  of  the  monument 
at  elevations  of  23°  20'  and  47°  30',  respectively? 

4.  A  mountain  peak  is  observed  from  the  base  and  top  of  a  tower 
200  ft.  high.  The  elevation  angles  being  25°  30'  and  23°  15',  respec- 
tively, compute  the  height  of  the  mountain  above  the  base  of  the  tower. 

5.  From  a  point  in  the  street  between  two  buildings  the  elevation 
angles  of  the  tops  of  the  buildings  are  30°  and  60°.     On  moving  across 


APPLICATIONS.  55 

the  street  20  ft.  toward  the  first  building  the  elevation  angles  are  found 
to  be  each  45°.  Find  the  width  of  the  street  and  the  height  of  each 
building. 

6.  From  the  peak  of  a  mountain  two  towns  are  observed  due  south. 
The  first  is  seen  at  a  depression  of  48°  40',  and  the  second,  8  miles  farther 
away  and  in  the  same  horizontal  plane,  at  a  depression  of  20°  50'.  What 
is  the  height  of  the  mountain  above  the  plane  ? 

7.  A  building  145  ft.  long  is  observed  from  a  point  directly  in  front 
of  one  corner.  The  length  of  the  building  subtends  tan-i  3,  and  the 
height  tan-i  2.     Find  the  height. 

8.  An  inaccessible  object  is  observed  to  lie  due  N.E.  After  the  ob- 
sen^er  has  moved  S.E.  2  miles,  the  object  lies  N.N.E.  Find  the  distance 
of  the  object  from  each  point  of  observation. 

9.  Assuming  the  earth  to  be  a  sphere  with  a  radius  of  3963  miles, 
find  the  height  of  a  lighthouse  just  visible  from  a  point  15  miles  distp,nt 
at  sea. 

10.  The  angle  of  elevation  of  a  tower  120  ft.  high  due  north  of  an 
observer  was  35° ;  what  will  be  its  angle  of  elevation  from  a  point  due 
west  from  the  first  point  of  observation  250  ft.  ?  Also  the  distance  of 
the  observer  from  the  base  of  the  tower  in  each  position  ? 

11.  A  railway  5  miles  long  has  a  uniform  grade  of  2°  30' ;  find  the  rise 
per  mile.     What  is  the  grade  when  the  road  rises  70  ft.  in  one  mile? 

(The  grade  depends  on  the  sine  of  the  angle.) 

12.  The  foot  of  a  ladder  is  in  the  street  at  a  point  30  ft.  from  the 
line  of  a  building,  and  just  reaches  a  window  22^  ft.  above  the  ground. 
By  turning  the  ladder  over  it  just  reaches  a  window  36  ft.  above  the 
ground  on  the  other  side  of  the  street.     Find  the  breadth  of  the  street. 

13.  From  a  point  200  ft.  from  the  base  of  the  Forefathers'  monument 
at  Plymouth,  the  base  and  summit  of  the  statue  of  Faith  are  at  an  eleva- 
tion of  12°  40'  48"  and  22°  2'  53",  respectively ;  find  the  height  of  the 
statue  and  of  the  pedestal  on  which  it  stands. 

14.  At  a  distance  of  100  ft.  measured  in  a  horizontal  plane  from  the 
foot  of  a  tower,  a  flagstaff  standing  on  the  top  of  the  tower  subtends  an 
angle  of  8°,  while  the  tower  subtends  an  angle  of  42°  20'.  Find  the 
length  of  the  flagstaff. 

15.  The  length  of  a  string  attached  to  a  kite. is  300  ft.  The  kite's 
elevation  is  56°  6'.     Find  the  height  of  the  kite. 

16.  From  two  rocks  at  sea  level,  50  ft.  apart,  the  top  of  a  cliff  is  ob- 
served in  the  same  vertical  plane  with  the  rocks.  The  angles  of  eleva- 
tion of  the  cliff  from  the  two  rocks  are  24°  40'  and  32°  30'.  What  is  the 
height  of  the  cliff  above  the  sea  ? 


CHAPTER  VI. 

GENERAL  FORMULA  —  TRIGONOMETRIC  EQUATIONS 
AND   IDENTITIES. 

49.  Thus  far  functions  of  single  angles  only  have  been 
considered.  Relations  will  now  be  developed  to  express 
functions  of  angles  which  are  sums,  differences,  multiples, 
or  sub-multiples  of  single  angles  in -terms  of  the  functions 
of  the  single  angles  from  which  they  are  formed. 

First  it  will  be  shown  that, 

sin  (a  ±  p)  =  sin  a  cos  p  ±  cos  a  sin  p, 
cos  (a  ±  p)  =  cos  a  cos  p  T  sin  a  sinpe 
tan  g  ±  tan  p 
1  T  tan  a  tan  p 

The  following  cases  must  be  considered : 

1.  a,  y8,  a  +  yS  acute  angles. 

2.  a,  y8,  acute,  but  a  +  yS  an  obtuse  angle. 

3.  Either  a,  or  y8,  or  both,  of  any  magnitude,  positive  or 
negative. 

The  figures  apply  to  cases  1  and  2. 


tan  (a  ±  p) 


Let  the  terminal  line  revolve  through  the  angle  «,  and 
then  through  the  angle  ^,  to  the  position  OB,  so  that  angle 

56 


GENERAL  FORMULA.  57 

XOB  =  a-{- 13.  Through  any  point  F  in  OB  draw  perpen- 
diculars to  the  sides  of  a,  DP  and  (7P,  and  through  C  draw 
a  perpendicular  and  a  parallel  to  OX,  MO  and  JVC, 

Then  the  angle  QCA  =  a  (why?),  and  CNP  is  the  triangle 
of  reference  for  angle  QCF  =  90°  +  a. 

CNP  is  sometimes  treated  as  the  triangle  of  reference  for  angle  CPN. 
The  fallacy  of  this  appears  when  we  develop  cos  («  +  ^),  in  which  PC 
would  be  treated  as  both  plus  and  minus. 

Now     sm(«  +  ^)=sinXO£  =  |^  =  ^+^, 

or  expressiftg  in  trigonometric  ratios, 

^MC    00^.  NP     CP_ 
OC'  OP     CP'  OP 
=  sin  a  cos  ^  +  sin  (90°  +  a)  sin  ff. 

Hence,  since  sin  (90°  4-  a)  =  cos  a,  we  have 

sin  (a  4-  yQ)  =  sin  a  cos  ^  +  cos  a  sin  ff. 
In  like  manner 

cos(«  +  ^)  =  cosXO^  =  — =  — +  -^ 

or  expressing  in  trigonometric  ratios, 

OM    00      ON    OP 


00     OP     OP    OP 

=  cos  a  cos  P  +  cos  (90°  +  a)  sin  yS. 

And  since  cos  (90°  +  a)  =  —  sin  a,  we  have 

cos  (a  +  y8)  =  cos  a  cos  yS  —  sin  a  sin  y5. 

It  will  be  noted  that  the  wording  of  the  demonstration  ap- 
plies to  both  figures,  the  only  difference  being  that  when  a  +  /3 
is  obtuse  OD  is  negative.      ON  is  negative  in  each  figure. 

50.  In  the  case,  when  a,  or  /3,  or  both,  are  of  any  magni- 
tude, positive  or  negative,  figures  may  be  constructed  as 
before  described  by  drawing  through  any  point  in  the  terminal 
line  of  P  a  perpendicular  to  each  side  of  a,  and  through  the  foot 
of  the  perpendicular  on  the  terminal  line  of  a  a  perpendicular 
and  a  parallel  to  the  initial  line  of  a.     Noting  negative  lines, 


68  PLANE   TRIGONOMETRY. 

the  demonstrations  already  given  will  be  found  to  apply  for 
all  values  of  a  and  y8. 

To  make  the  proof  complete  by  this  method  would  require  an  unlim- 
ited number  of  figures,  e.g.  we  might  take  a  obtuse,  both  a  and  (i  obtuse, 
either  or  both  greater  than  180°,  or  than  360°,  or  negative  angles,  etc. 

Instead  of  this,  however,  the  generality  of  the  proposition 
is  more  readily  shown  algebraically,  as  follows : 

Let  a^  =  90°  +  a  be  any  obtuse  angle,  and  a,  yS,  acute 
angles. 

Then  ^ 

sin  Qa!  +  iS)  =  sin  (90°  +  a  +  yS)  =  cos  (a  +  ^S) 
=  cos  a  cos  /3  —  sin  a  sin  y8 

=  sin  (90°  +  a)  cosy8  +  cos  (90°  +  a)  sinyS(why?) 
=  sin  a'  cos  /3  +  cos  ct'  sin  y(3. 

In  like  manner,  considering  any  obtuse  angle  ^'  =  90°  +  yS, 
it  can  be  shown  that 

sin  (a'  +  yS')  =  sin  ex!  cos  y8'  +  cos  aJ  sin/3^ 

Show  that  cos  (a'  +  ^8')  =  cos  a'  cos  fi'  —  sin  a'  sin  /3^ 

By  further  substitutions,  e.g.  a"  =  90°  ±  a',  0"  =  90°  ±  yS^ 
etc.,  it  is  clear  that  the  above  relations  hold  for  all  values, 
positive  or  negative,  of  the  angles  a  and  yS. 

Since  a  and  0  may  have  any  values,  we  may  put  —  yS  for  y8, 
and  sin(a+  [— yS]) 

=  sin  (a  —  yS)  =  sin  a  cos  (  —  y5)  +  cos  a  sin  (  —  yS) 

=  sin  a  cos  yS  —  cos  a  sin  yS  (why  ?) . 

Also  cos  (a  —  I3}=  cos  a  cos(  —  yQ)  —  sin  a  sin  (  —  /3) 

=  cos  a  cos  yS  +  sin  a  sin  /3, 
Finally, 

tan  r    +  iS^  —  ^"^  C^  ^  )^)  _  sin  ct  cos  y9  ±  cos  ct  sin  /S 
cos(a±/3)      cosctcosyS^  sinasinyS 
sinctcosy8     cos«siny8 
cos  a  cos  /S     cos  a  cos  yS       tan  a  ±  tan  y8 


cosctcosy^     sin  a  sin  y8      1  qp  tan  a  tan  y8 
cos  a  cos  /3     cos  a  cos  yS 


EXAMPLES.  59 

ORAL  WORK. 

By  the  above  formulae  develop : 

1.  sin  (2A  +SB).  7.  sin 90°  =  sin (45°  +  45°). 

2.  cos  (90° -5).  8.  cos  90°. 

3.  tan  (45°  +  <^).  9.  tan  90°. 

4.  sin  2  yl  =  sin  (A  +  A),  10.  sin  (90°  +  /?  +  y). 

5.  cos  2^.  11.  cos  (270°  -  m  -  n). 

6.  tan  (180°  +  C).  12.  tan  (90°  +  m  +  n). 

Ex.  1.   Find  sin  75°. 

sin  75°  =  sin  (45°  +  30°)  =  sin  45°  cos  30°  +  cos  45°  sin  30° 

=  ^.^  +  ^.Ul±^  =  0.9659. 
y/2.     2        V2    2        2\/2 

2.   Find  tan  15°. 

tan  45°  -  tan  30° 


tan  15°  =  tan  (45°  -  30°) 


i-X 


1  +  tan  45°  tan  30° 


^  =  ^-^  =  2  -  V3  =  0.2679. 


1  + J_      V3  +  1 
V3 

3.  Prove  !iEM_£2iM=2. 

sin  A        cos  A 

Combining"    ^^^  ^ ^  ^^^ ^  —cos  3 ^  sin ^  _  sin  (3^4  —  A^ 
sin  A  cos  A  sin  A  cos  A 

_     sin  2  A      _  sin  (A  -\-  A)  _  sin  A  cos  A  +  cos  A  sin  A  _  g 
sin  A  cos  A       smA  cos  A  sin  A  cos  A 

4.  Prove  tan-^  a  +  tan-^  b  =  tan-^  -^-^ — 

1  —  ah 

Let  a  —  tan-^a,  /8  =  tan-i&,  y  =  tan-^  ^  "^    * 

Hence,  tan  a  =  a,  tan  (3  =  b,  tan  y  =  -^^^ -• 

Then  a  -\-  fi  =  y,  and  hence  tan  («  +  )8)  =  tany. 

Expanding,  tan  «  +  tan /?  ^  ^^^ 

1  —  tan  a  tan  yt? 


Substituting, 


g  +  6  __a_±A. 
1  -  a6      1  -  a6* 


60  PLANE   TRIGONOMETRY. 

EXAMPLES. 

1.  Find  cos  15°,  tan  75°. 

2.  Prove  cot  (a  ±  (3)  =  ^ot^^cot^Tl. 

,     ^    ^^"^       cot)8±cota 

3.  Prove  geometrically  sin  («  +  /?)  =  sin  a  cos  )8  +  cos  a  sin  j8, 

and  cos  (a  +  )8)  =  cos  a  cos  ^  —  sin  a  sin  j3, 
given  (a)  a  acute,  y8  obtuse ; 

(b)  a,  P,  obtuse  ; 

(c)  a,  /3,  either,  or  both,  negative  angles. 

4.  Prove  geometrically  tan  (a  +  B)  =  ^^^<^J^^^^P 

^  J         V        A-y      l-tanatan/3 

Verify  the  formula  by  assigning  values  to  a  and  fi,  and  finding  the 
values  of  the  functions  from  the  tables  of  natural  tangents. 

5.  Prove  cos  (a  +  jS)  cos  (a  —  ft)  =  cos^  a  —  sin*^  fi. 

6.  Show  that  tan  a  +  tan  j8  =  sin  (a  +  p\ 

cos  a  cos  p 

7.  Given  tan  a  =  i,  tan  )8  =  f ,  find  sin  (a  +  )8) 

8.  Given  sin  280°  =  s,  find  sin  170°. 

9.  If  a  =  67°  22',  jS  =  128°  40',  by  use  of  the  tables  of  natural  func- 
tions verify  the  formulae  on  page  56. 

Prove  tan-i  ^^  "^  ^  =  tan-V^+  tan-^Va. 


=tan-iV3. 


13.  If  a  +  /3  =  <o,  prove  cos^ «  +  cos^  j8  —  2  cos  a  cos  j8  cos  cu  =  sin*  w. 

14.  Solve  i  sin  ^  =  1  —  cos  0. 

15.  Prove  sin  (A  +  B)  cos  A  —  cos  (A  +  E)  sin  A  =  sin  J5. 

16.  Prove  cos  {A  +  B)  cos  {A-B)-\-  sin(^  +  B)  sin(^  -B)  =  cos  2  B. 

17.  Prove  sin  (2  a-  ft)  cos  {a -2  ft) 

-  cos  (2  a-  ft)  sin  (a  -  2  j8)  =  sin  (a  +  /S). 

18.  Prove  sin(n  — l)acos(n4-l)a  +  cos(n-l)asin(n  +  l)a=  sin2n«. 

19.  Prove  sm  (135°  -  0)  +  cos  (135°  +  ^)  =  0. 


1-V^ 

..  Prove  tan- 

b^/3 

2b-x 

xV3 

!.   Prove  sec~ 

I       ""         -  sin 

-lE. 

y/a^-x^ 

a 

ADDITION— SUBTRACTIOX   FORMULA.  61 

20.  Prove  1  -  tan^  « tan^  R  =  cos^  jg  -  si^^  ^. 

cos*  a  cos*  /? 

21.  Prove  t^°«  +  tan^  ^  j^^  „  ^^^  ^ 

cot  a  +  cot  p 


22. 


tan*  f ^  -  «U  l-2sin«cosci, 
\4         /     1  4-  2  sin  a  cos  a 


51.  The  following  formulae  are  very  important  and  should 
be  carefully  memorized.  They  enable  us  to  change  sums 
and  differences  to  products,  i.e.  to  displace  terms  by  factors. 

sine  +  siii<|»  =  2  sin^^cos^^, 
sine  -  sin<t>  =  2cos-^t_?sin— =-5, 
COS0  +  COS«|>  =  2cos-^^cos— ^5 
COS  0  -  cos<}>  =  -  2  sin  -i-^  sin  — ^• 

Since  sin  («  +  y8)  =  sin  «  cos  y8  +  cos  a  sin  /8, 

and  sin  («  —  )9)  =  sin  a  cos  y8  —  cos  a  sin  ^, 

then  sin  (a  +  y8)  +  sin  (a  —  /8)  =  2  sin  a  cos  ^9,         (1) 

and  sin  (a  +  /S)  —  sin  (a  —  y8)  =  2  cos  a  sin  ^.  (2) 

Also  since  cos  («  +  yS)  =  cos  a  cos  y8  —  sin  a  sin  yS, 

and  cos(a  — y8)=  cosacosy8+ sinasinyS, 

then  cos  (a  + /3)  +  cos  (a  — /3)  =  2  cos  a  cos  y8,         (3) 

and  cos(a  +  yS)— cos(a  — /3)=  —  2sinasin/S.      (4) 

Put  a-\-^  =  e 

and  a  — 13  =  cl> 

2a  =  0  +  <j>,  and  a  =  ^±-i, 

2/3  =  ^-^,  andyg  =  ^-=^. 

A 

Substituting  in  (1),  (2),  (3),  (4),  we  have  the  above 
formulae. 


62  PLANE   TRIGONOMETRY. 


EXAMPLES. 

1.  Prove  ?HL24±iHi|  =  tan  ^. 
cos  2  ^  +  cos  ^  2 

By  formulae  of  last  article  the  first  member  becomes 

2  sm  —  cos  - 

2         2  3^ 

=  tan 


o     3(9     e        2 

2  cos  —  cos  - 
2         2 

2    p  sin  ct  4-  2  sin  3  g  +  sin  5 ct  _  sin3  a 

sin  3  cc  +  2  sin  5  ct  +  sin  7  a     sin  5  a 

(sin  ct  4-  sin  5  g)  +  2  sin  3  (z   _  2  sin  3  a  cos  2  «  +  2  sin  3  ct 
(sin  3  ct  +  sin  7  a)  +  2  sin  5  a     2  sin  5  cc  cos  2  ct  +  2  sin  5  a 

_  (cos  2  (^  +  1)  sin  3  ot  _  sin  3  a 
(cos  2  a  +  1)  sin  5  ct     sin  5  a 

3.  Prove  ^^"  ^^^  -  2  g)Hh  sin  (4  ^  -  2  ^)  ^  ^^^ 

cos  (4^ -25)+ cos  (45 -2^)  ^  ^ 

o-    4^-25+45-2^        4^-25-45+2^ 

2  sin cos ■ 

2 2 

o        4yl -25  +  45-2^1        4^-25-45  +  2^ 

2  cos ;^ COS ■ 

2  2 

=  ^-niM+^  =  tan(^+5). 
COS  (.1+5)  ^  ^ 

4.  Prove  sin  50°  -  sin  70°  +  sin  10°  =  0. 

2  cos  ^^°  "^  '^^°  sill  ^^'^  ~  ^^"^  =  2  cos  60°  sin  (  -  10°)  =  -  sin  10°. 
2.  2  '  ^ 

5.  Prove  ^^^^""^"^^-^^^^^^^"'^^  +  ^^^^^^Q"^Q^=cot6(^cot5tt. 

sin4asin3ct— sin2()isiri  5  a  +  sin4otsin7a 

By  (3)  and  (4),  p.  61, 

cos  5  ct  +  cos  a  —  cos  9a  —  cos  5  ot  +  cos  11  ct  +  cos  9  ct 
cos  a  —  cos  7  a  —  cos  3  a  +  cos  7  a  +  cos 'S  a  —  cos  11  a 

cos  a  +  cos  11a     2  cos  6  a  cos  5  a     ^^4.  «  ^  «^f  k  « 
= ! =  — —  =  cot  6  a  cot  0  a. 

cos  a  —  cos  11a     2  sin  6  a  sin  5  a 

ORAL  WORK. 
By  the  formulae  of  Art.  51  transform : 

6.  cos  5  a  +  cos  a.  8.  2  sin  3  d  cos  $. 

7.  cos  a  —  cos  5  a.  9.  sin  2  a  —  sin  4  a. 


FUNCTIONS  OF  THE  DOUBLE  ANGLE.       63 

10.  cos  9^  cos  2^.  16.   cos(30°+2<^)sin(30°-</)). 

Q 

11.  sin  $  +  sin  -.  17.   sin  (2  r  +  s)  +  sin  (2  r  -  s). 

12.  sin  75°  sin  15°.  18.  cos  (2  )8  -  a)  -  cos  3  a. 

13.  cos7i>-cos2;7.  19.   sin  36°  +  sin  54°. 

14.  cos(2o  +  3o)sin(2p-3o). 

^  ^        ^^       ^  -^        ^^  20.   cos  60°  + cos  20°. 

15      •    ?i        '    L 

'  ^^^2  ~^"^2'  21.   sin  30°  + cos  30°. 

•      Prove:  22.   ?HL^^±^  =  tan«4J?cot^^:^. 

sin  a  —  sin  y8  2  2 

23    cos  «  +  cos  ^  ^  cotgL±^cot^^-^. 
cos  /?  —  cos  a  2  2 

2^^   sin^  +  siny^^^^^+j^^ 
cos  X  +  COS  y  2 

25.  sin  a:- sin  y  ^  _  ^ot^i^^. 
cos  a:  —  cos  y  2 

26.  cos  55°  +  sin  25°  =  sin  85°. 

Simplify:        27.   sin^  +  sin  2  B  +  sin  3  ^^ 
cos  B  +  COS  2B  +COS  3  5 

23    sin  C  -  sin  4  C  +  sin  7  C  -  sin  10  C 
cos  C  —  COS  4  C  +  cos  7  C  —  cos  10  C 

52.  Functions  of  an  angle  in  terms  of  those  of  the  half  angle. 
If  in  sin  (a  +  /3)  =  sin  a  cos  yS  +  cos  a  sin  jS,  a  =  j3, 
then       sin  (a  +  a)  =  sin  2  a  =  2  sin  a  cos  a. 
In  like  manner 

cos  (a  +  a)  =  COS  2  a  =  cos^  a  -  sin^  a 

=  2  cos^  a  - 1  ^ 

=  l-2sin»a; 

and  tan  2  a  = 


1  -  tan*  a 


64  PLANE   TRIGONOMETRY. 

ORAL  WORK. 

Ex.    Express  in  terms  of  functions  of  half  the  given  angles : 

1.  sin  4  a.  4.   cos  a:.  6.   sm(2p  —  q). 

2.  cos3».  .    Q  7.   cos  (30°  +  2  6). 

5.   sm^. 

3.  tan5^  2  8.    sin  (a;  +  ?/). 

9.   From  the  functions  of  30°  find  those  of  60° ;  from  the  functions  of 
45°,  those  of  90°. 

53.  Functions  of  an  angle  in  terms  ^  those  of  twice  the  angle. 

By  Art.  52,     cos  a  =  1  -  2  sin2  ^  =  2  cos2^  -  1. 

-^  2  2 


2sin^|  = 

=  1  —  cos  «, 

and 

2cos2^ 

«-r 

.    a 

sm- 

a 
cos- 

-COS  a 
2       ' 

-^ 

...  u.|= 

^^1-cosa 
^  1  +  COS  a 

1=^4 


1  +  cos  a 


Explain  the  significance  of  the  ±  sign  before  the  radicals. 
Express  in  terms  of  the  double  angle  the  functions  of 
120°;  50°;  90°,  with  proper  signs  prefixed. 

Ex.  1.   Express  in  terms  of  functions  of  twice  the  given  angles  each 
of  the  functions  in  Examples  1-8  above. 

2.  From  the  functions  of  45°  find  those  of  22°  30' ;  from  the  functions 
of  36°,  those  of  18°  (see  tables  of  natural  functions). 

3.  Find  the  corresponding  functions  of  twice  and  of  half  each  of  the 
following  angles,  and  verify  results  by  the  tables  of  natural  functions : 

Given  sin  26°  42'  =  0.4493, 

tan  62°  24' =  1.9128, 

cos  21°  34'  =  0.9300. 


-4 


4.   Prove  tan-iA/^— -^^^  =  ?.  5.  2  tan-i a;  =  tan-J    ^^ 


+  cos  X     2  1  —  x^ 


EXAMPLES.  65 

6.  Ji  Af  B,  C  are  angles  of  a  triangle,  prove 

sin ^  +  sin  C  +  sinjB  =  4  cos  —  sin  —  sin  -^ 

2        2        2 

7.  K  cos2 a  +  cos2 2a  +  cos^ 3 a  =  1,  then 

cos  a  cos  2  a  cos  3  a  =  0. 

8.  Prove  cot ^  —  cot  2 ^  =  esc  2 ^. 

2 


tan 

9.  Prove 

tan 


(H) 
(M) 


1  —  tan  2 
2 

l  +  tan|j 


10  tang       _  - 2  sin  ^ ^ 

tan  (a  +  <^)  sin  (2  a  +  <^)  +  sin  <^ 


U.  lfv  =  t2^n-i2^I±^'  +  ^^^^^\  prove  2:2  =  sin  2y. 


12.  Prove  tan-i  Vl  +  x^-  1      ^an-i  -1^  =  g  tan-i x. 

X  1  -  a;2     2 

13.  If  y  =  sin-i  -     ^         prove  x  =  tan  w. 

14.  Prove  cos2  «  +  cos2  j8  -  1  =  cos  (a  +  fS)  cos  (a  -  ^). 

15.  Prove  V(cos  a  -  cos  ^)  2  -f-  (sin  a  -  sin  ^8)2  =  2  sin  £LZL^. 


16.  Prove  sin-i  V-^—  =  tan-i  J-  =  -  cos-i  ^^ 
^a+a;  ^a     2  a  + 


17.  Prove  cos^  0  -  cos^  <^  =  sin  (<^  +  6)  sin  (<l>  -  0). 

18.  Prove  tan  ^  +  tan  (A  +  120°)  +  tan  (A  -  120°)  =  3  tan  3  ^. 

19.  Prove  tan  a  —  tan  -  =  tan  -  sec  a. 

2  2 

20.  3tan-ia  =  tan-i^"~^^ 

1  -  3  a2 

21.  cos2  3  A  (tan2  3  ^  -  tan2 ^)  =  8  sin2^  cos  2  ^. 

22.  1  +  cos  2  (^  -  B)  cos  2 5  =  cos2^  +  cos2  (A  -2 B). 


23.   cot2fE  +  ^U2csc26l- 
\4      2/     2csc2^  + 


sec  0 
seed 


66  PLANE  TRIGONOMETRY. 

TRIGONOMETRIC  EQUATIONS  AND  IDENTITIES. 
54.   Identities.     It  was  shown  in  Chapter  I  that 
sin2  e  +  cos2  0  =  1 
is  true  for  all  values  of  0,  and  in  Chapter  VI,  that 

sin  (a  +  /3)  =  sin  a  cos  /3  +  cos  a  sin  jS 
is  true  for  all  values  of  a  and  /3.     It  may  be  shown  that 


sin  2  A       "^ 


=  tan-4 


1  +  cos  2  J. 

is  true  for  all  values  of  A^  thus : 

sin  2  A  _  2  sin  A  cos  A  (by  trigonometric  transf  orma- 
1  +  cos2J.~l  +  2cos2J.-l       tion) 

= J  (by  algebraic  transformation  J 

=  tsinA  (by  trigonometric  transformation). 

Such  expressions  are  called  trigonometric  identities.  They 
are  true  for  all  values  of  the  angles  involved. 

55.  Equations.     The  expression 

2  cos^  a  —  3  cos  a +  1  =  0 

is  true  for  but  two  values  of  cos  a,  viz.  cos  a=  ^  and  1,  i.e. 
the  expression  is  true  for  a  =  0°,  60°,  300°,  and  for  no  other 
positive  angles  less  than  360°.  Such  expressions  are  called 
trigonometric  equations.  They  are  true  only  for  particular 
values  of  the  angles  involved. 

56.  Method  of  attack.  The  transformations  necessary  at 
any  step  in  the  proof  of  identities,  or  the  solution  of  equa- 
tions, are  either  trigonometric^  or  algebraic;  i.e.  in  prov- 
ing an  identity,  or  solving  an  equation,  the  student  must 
choose  at  each  step  to  apply  either  some  principles  of  algebra, 
or  some  trigonometric  relations.  If  at  any  step  no  algebraic 
operation  seems  advantageous,  then  usually  the  expression 


METHOD  OF  ATTACK.  67 

should  be  simplified  by  endeavoring  to  state  the  different 
functions  involved  in  terms  of  a  single  function  of  the  angle, 
or  if  there  are  multiple  angles^  to  reduce  all  to  functions  of  a 
single  angle. 

Algebraic 


Transformations 


Trigonometric,     f  Single  function 
to  change  to  a  1  Single  angle 


No  other  transformations  are  needed,  and  the  student  will 
be  greatly  assisted  by  remembering  that  the  ready  solution 
of  a  trigonometric  problem  consists  in  wisely  choosing  at 
each  step  between  the  possible  algebraic  and  trigonometric 
transformations.  Problems  involving  trigonometric  func- 
tions will  in  general  be  simplified  by  expressing  them  entirely 
in  terms  of  sine  and  cosine. 

EXAMPLES. 

T     -D  sin  3  ^      cos  3  ^      n 

1.  Prove  — : — : —  =  2. 

smA         cos  A 


By  algebra. 


sin  3^      cos  3  ^  _  sin  3  ^  cos  ^  —  cos  3  J  sin  A 
sin  A        cos  A  sin  A  cos  A 


...  ., sm  (3  ^  —  ^ )         sm  2  ^ 

by  trigonometry,  =  — r^^- 7^  =  -: — ;; 7 

sin  ^  cos  A        smA  cos  A 

_  2  sin  ^  cos  A  _  n 
sin  A  cos  A 
Or,  by  trigonometry, 

sin  SA      cos  3  ^  _  3  sin  ^  -  4  sin»  A      4  cos^  A  -  3  cos  ^ 
sin -4        cos -4  sin^  cos  J. 

by  algebra,  =3-4  sin2  A  —  4:  cos^  ^  +  3 

=  6  -  4(sinM  +  cosM)=  2. 

sec  8  ^  -  1      tan  8  ^ 


2.   Prove 


sec  4  ^  -  1      tan  2  ^ 


No  algebraic  operation  simplifies.  Two  trigonometric  changes  are 
needed.  1.  To  change  the  functions  to  a  single  function,  sine  or  cosine. 
2.  To  change  the  angles  to  a  single  angle,  8 yl,  4 ^,  or  2 ^. 


68  PLANE  TRIGONOMETRY. 

By  trigonometry  and  algebra, 

1  -  cos  8  ^     sin  8  ^ 


P  cos  8  ^      _  cos  8  e       _      tavx  <f  ^  ^ 

^   l-cos4^-sin2|>      -     ^^^p.  \ 

cos  4^         cos  2^  iiV^^-v^^ 

K^  oin.nK,.o  cos  4 ^(1  -  cos  8  0)      sin  8  6  cos^.  cr  _  <^  ^^  _ 

by  algebra,  — ^^ -^ — ^  = ♦    o^       *  '^  "=       ii-^"*-  •?  /5«r~ 

1  —  cos 4 0  sm2^  /. &^-»^  _j_r 

by  trigonometry,  /^  .        ^ 

COS  4  ^(1  -  1  +  2  sin2  4  ^)  Z^  sin  4  ^  cos  4  ^  cos  2  g .    '"'  iT&II^T^r^ 
l-l  +  2sin22^    /<"  sin2^  '        .     ,  -       .      « 

by  algebra,  /^^^  =  2  cos  2 ^ ;  ~  C«^ «r^ .  ^!.ui-^&c. 

ysin  2  0  -        /L 

/  _,  S»Vt,  frO"  C4H7  2.  £ 

and  X      sin  40  =  2  sin  2  0  cos  2ft  —     Co^t^  '  "sT^ItI 

which  is  a  trigonometric  identity. 

J 

3.  Solve  2  cos2  0  +  3  sin  0  =  0.  | 

By  trigonometry,      2(1  -  sin^  0)  +  3  sin  0  =  0,  3 

a  quadratic  equation  in  sin  0.  -, 

"I 

By  algebra,  2sin20  -  3sin0  -  2  =  0,  ] 

and  (sin0-2)(2sin0  +  l)  =  O.  ^ 

.*.  sin  0  =  2,  or  —  ^.     Verify. 

The  value  2  must  be  rejected.    Why?  ' 

.-.  0  =  210°,  and  330°  are  the  only  positive  values  less  than  360°  that  ■ 

satisfy  the  equation. 

■  '\ 

4.  Solve  sec  0  —  tan  0  =  2.  i 
Here  tan  0  =  —  0.75,  .-.  from  the  tables  of  natural  functions,  *, 

0  =  143°  7'  48",  or  323°  V  48".  j 

Find  sec  0,  and  verify.  I 

5.  Solve  2  sin  0  sin  3  0  -  sin^  2  0  =  0.  j 
By  trigonometry,      cos  2  0  —  cos  4  0  —  sin^  2  0  =  0,  I 

also                    cos20-cos2  20  +  sin22  0-sin220  =  O.  j 

By  algebra,                  cos  2  0(1  -  cos  2  0)  =  0.  | 

.-.  cos  2  0  =  0  or  1,  i 

and                                 2  0  =  90°,  270°,  0°,  or  360°,  j 

*  whence                            0  =  45°,  135°,  0°,  or  180°.    Verify.  i 


TRIGONOMETRIC  EQUATIONS.  69 

Or,  by  trigonometry, 

2  sin  ^(3  sin  ^  -  4  sin^  ^)  -  4  sin2  0  cos^  d  =  0 ; 
by  trigonometry  and  algebra, 

6  sin2  ^  -  8  sin*  ^  -  4  sin2  ^  +  4  sin4  ^  =  0; 
by  algebra,  2  sin^  ^  -  4  sin'*  ^  =  0, 

and  2  sin^  ^(1  -  2  sin2  $)  =  0. 

.-.  sin  ^  =  0,  or  ±  V|, 
and  6  =  0°,  180°,  45°,  135°,  225°,  or  315°. 

The  last  two  values  do  not  appear  in  the  first  solution,  because  only 
angles  less  than  360°  are  considered,  and  the  solution  there  gave  values 
of  2  0,  which  in  the  last  two  cases  would  be  450°  and  630°. 

Solve :       6.  tan  ^  =  cot  ^.  8.  2  cos  2  ^  -  2  sin  ^  =  1. 

7.  sin^  ^  +  cos  ^  =  1.  9.   sin  2  d  cos  0  =  sin  0, 

Prove:     10.  2cot2^  =  cot^  —  tan^. 

11.  cos  2  a:  +  cos  2  y  =  2  cos  (x  +  y)  cos  (x  —  y). 

12.  (cos  a  +  sin  ay  =  1  +  sin  2  a. 

57.   Simultaneous  trigonometric  equations. 
13.   Solve  cos  (x  -h  y)+  cos  (x  -  y)  =  2, 

sin  -  +  sin  ^  =  0. 
2  2 

By  trigonometry, 

cos  x  cos  y  —  sin  x  smy  +  cos  x  cos  y  +  sin  ar  sin  y  =  2, 


so  that 

cos  a:  cos  y  =  1 ; 

also, 

^ 

-cosar     Jl-cosy_Q 

2^2 

and 

.'.  cos  X  =  COS  y. 

Substituting, 

cos2  a:  =  1, 
COS  a;  =  ±  1. 
.-.  X  =  0°,  or  180°, 

and 

y  =  ar  =  0°,  or  180°.    Verify, 

70  PLANE  TKIGONOMETRY 

14.  Solve  for  R  and  F. 

W  —  Fsini  —  R  cos i  =  0, 

W  +  F  cos  i  —  R  sin  i  =  0. 
To  eliminate  F, 

Wcos  i  —  Fshi  i  cos  i  —  R  cos^i  =  0, 

W  sin  i  +  jPcos  i  sin  i  —  R  sin^  i  =  0. 
Adding,  TF(sin  i  +  cos  i)  —  R(sm^  i  +  cos^  i)  =  0. 

.-.  R  =  W(sm  i  +  cos  i). 
Substituting,    W  —  Fsini  —  W((sin  i  +  cos  z)cos  i  =  0 

ET  _  W  —  Ty(sin  i  +  cos  i)  cos  z 
'  *  •  sin  i 

Jl  W  =  S  tons,  and  i  =  22°  30',  compute  F  and  jR. 

i2  =  3(0.3827  +  0.9239)=  3.9198. 

F  ^  3  -  3(0.3827  +  0.9239)0.9239  ^_iqoa 

0.3827  ■       * 

Solve : 

15.  472  cot  e  -  263  cot  <^  =  490,  307  cot  6  -  379  cot  <^  =  0. 

16.  sin  2  a:  +  1  =  cos  a;  +  2  sin  a;. 

17.  cos2  e  +  sin  ^  =  1. 

18.  If  2;^(cos2^-sin2^)-2asin^cos^  +  26sin^cos^  =  0,   prove 

^  =  itan-i-?-^. 
a  —  b 

Prove : 

19.  tan  y  =(1  -{■  sec  y)  tan  ^' 

20.  2  cot-i  X  =  csc-i  ^  "^  ^^- 

2a: 

21.  sin(<^  +  45°)  +  sin  (<^  +  135°)  =  V2  cos  <^. 
22      cos  V  +  cos  3  y    _  1 


cos  3  y  +  cos  5v      2  cos  2  w  —  sec  2 1' 

23.  cos  3  a:  —  sin  3  a:  =  (cos  x  +  sin  a;)  (1  —  2  sin  2  x). 
Solve : 

24.  sin  2  ^  +  sin  ^  =  cos  2  ^  +  cos  6. 

25.  4  cos(^  +  60°)  -  V2  =  Ve  -  4 cos  (^  +  30°). 

26.  tan  2  ^  =  tan  0-1. 

27.  cos  ^  +  cos  2  ^  +  cos  3  ^  =  0. 


TRIGONOMETRIC   EQUATIONS.  71 

28.  sin  2  a;  +  V3  cos  2  a;  =  1. 

29.  3  tan'-^jo  +  8  cos^p  =  7. 

30.  Determine  for  what  relative  values  of  P  and  W  the  following 
equation  is  true : 

cos2^-  — cos^-i=0. 
2      W       2     2 

31.  Compute  N  from  the  equation  iV+ -—  cos  a  —  —  sin  a  —  TV  cos  a  =  0, 

o  o 

when  W  =  2000  pounds  and  a  satisfies  the  equation  2  sin  ct  =  1  +  cos  a. 

32.  sin  9  —  tan  <j>(cos  6  +  sin  0)  =  cos  0,  sin  ^  —  tan  ^  cos  ^  =  1. 
Prove : 

33.  coi{t  +  15°)  -  tan (t  -  15°)  =     ^  ^^^  ^  < 


2  sin  2  <  +  1 
34.   sin-i  f  —  sin~i  y\  =  sin"^  ^. 


35.   tan(^  +  ^UV^+^"^^. 
U      2/      >l-sino> 


36.  2  sin-i  i  =  cos-i  1. 

37.  If  sin  ^   is  a  geometric  mean  between  sin  B  and  cos  B,  prove 
cos  2^  =2sin(45  -  5) cos (45  +  B). 

38.  Prove     sin  (a  +  y8  +  y)  =  sin  a  cos  y8  cos  y  +  cos  «  sin  ^  cos  y 

+  cos  «  cos  y8  sin  y  —  sin  a  sin  j8  sin  y. 
Also  find  cos(a  +  /S  +  y). 

39.  Prove  tan((.  + /3  +  y)  :^  ^^"  ^  +  ^^"  ^  +  ^^^  V  "  *^^  ^  *^"  ^  ^^^  ^. 

1  —tan  a  tan  ^— tan  )8  tan  y— tan  y  tan  « 

If  a,  jS,  and  y  are  angles  of  a  triangle,  prove 

40.  tan  a  +  tan  ^  +  tan  y  =  tan  «  tan  y8  tan  y. 

41.  cot-  +  cot^4-cot^  =  cot-cot^cotX 

2  2  2  2        2        2 

If  a  +  /8  +  y  =  90°,  prove 

42.  tan  a  tan  ^  +  tan  ^  tan  y  +  tan  y  tan  a  =  1. 
Prove : 

43.  sin  na  =  2  sin  (n  —  1)  a  cos  a  —  sin  (n  —  2)a. 

44.  cos  na  =  2  cos  (n  —  1)  a  cos  a  —  cos  (n  —  2)a. 

45.  tann«=   tan  (n  -  1)«  +  tan  «  , 

1  —  tan  (n  —  1)  a  tan  a 


CHAPTER  VII. 

TRIANGLES. 

58.  In  geometry  it  has  been  shown  that  a  triangle  is 
determined,  except  in  the  ambiguous  case,  if  there  are  given 
any  three  independent  pai^s,  as  follows : 

I.   Two  angles  and  a  side. 
II.    Two  sides  and  an  angle, 
(<x)  the  angle  being  included  by  the  given  sides, 
(5)  the  angle  being  opposite  one  of  the  given  sides  (am- 
biguous case). 
III.    Three  sides. 

The  angles  of  a  triangle  are  not  three  independent  parts,  since  they 
are  connected  by  the  relation  A  ■\-  B  +  C  =  180°. 

The  three  angles  of  a  triangle  will  be  designated  A^  B,  O, 
the  sides  opposite,  a,  b,  c. 

But  the  principles  of  geometry  do  not  enable  us  to  compute 
the  unknown  parts.  This  is  accomplished  by  the  following 
laws  of  trigonometry : 

I.   Law  of  Sines,  5«L4  =  !ilL?  =  !iH-2. 

a  0  c 

II.    Law  of  Tangents,   tan :^ (^  -  ^)  ^  a-| 
•^  ^  tani-(^  +  ^)      a-\-b 

III.    Law  of  Cosines,  cos  A  =  — \^ ,  ~"  ^  ,  etc. 

2oc 

59.  Law  of  Sines.  In  any  triangle  the  sides  are  propor- 
tional to  the  sines  of  the  angles  opposite. 

Let  ABQ  be  any  triangle,  p  the  perpendicular  from  B 
on  h.     In  I  (Fig.  34),  C  is  an  acute,  in  II,  an  obtuse,  in  III, 

72 


LAW  OF   SINES  — OF   TANGENTS. 


73 


a  right  angle.  The  demonstration  applies  to  each  triangle, 
but  in  II,  ^mACB=^\nDOB  (why?);  in  III,  sinC=l 
(why?). 


P 
Now  sin  A  —  —'>  ,'.  v  =  c  sin  A, 

c  ^ 

P 

sin  C=  —1  .'.  p  =  a  sin  (7. 

Equating  values  of  ^,  cs>vn.A  =  a  sin  (7, 

sin  A      sin  C 

or,  = 

a  c 

By  dropping  a  perpendicular  from  A^  or  (7,  on  a,  or  (?,  show 

sin  B     sin  C        sin  ^      sin  B 
,  or 


whence 


he  a 

sin  A      sin  B     sin  (7 


6    ' 


60.  Law  of  Tangents.  The  tangent  of  half  the  difference 
of  two  angles  of  a  triangle  is  to  the  tangent  of  half  their  sum, 
as  the  difference  of  the  sides  opposite  is  to  their  sum, 

a  __  sin  A 
h~ 


By  Art.  59, 


or 


sin  j5 
By  composition  and  division, 

g  -  5  ^  sin  J.  —  sin  ^  ^  2  cos  IQA  +  B)  sin  i  ( J.  -  ^) 
a  4-  ^  ~  sin  ^  +  sill  ^  ~  2  sin  i^  ( Jl  +  ^)  cos  ^(^A  —  B^ 
^tan^(J.-jg), 
tan|-(^  +  J5)' 

tan ^{A  —  B)  _a—b 
tani(^H-5)~a  +  6' 


74 


PLANE   TRIGONOMETRY. 


61.  Law  of  Cosines.  The  cosine  of  any  angle  of  a  triangle 
is  equal  to  the  quotient  of  the  sum  of  the  squares  of  the  adjacent 
sides  less  the  square  of  the  opposite  side,  divided  hy  twice  the 
product  of  the  adjacent  sides. 


In  each  figure  a^=p'^-\-DC^ 

=  c^-AD^  +  (h-AI>y 

(in  Fig.  34,  II,  DC  is  negative ;  in  III,  zero) 

=  c2  -  AB^  ^P-2b'Al)  +  Aiy^ 

=  h'^-\-c^-2h-AD. 
But 

AD  =  ccosA,     .'.  a^  =  P-\-(^-2boGOsA; 

\osA^'I±^-Z^. 
2  be 

Prove  that  cos B  =  ^^  +  g^-^^ 

2aG      ' 


and 


2ab 


62.  Though  these  formulae  may  be  used  for  the  solution 
of  the  triangle,  they  are  not  adapted  to  the  use  of  loga- 
rithms (why?).     Hence  we  derive  the  following: 

Since     cos  J.  =  2  cos^  -4-1  =  1-2  sin2^, 
2  2 

we  have 

nA  A 

2  cos^^  =  1  +  cos  A,  and  2  sin^-  =  1  -  cos -4. 
z  2 


LAW  OF  COSINES. 


75 


From  the  latter 

0.0-^1         524-c2_^2        25^-62-^2  _|.^2 

^^^"2=^-        2hc        = Ihc 

2  be  2  be 

Let     a  +  5  +  c  =  2s,  then  a-\-b  —  e=a  +  b-}-c—2c=2s—2e; 
i.e.  a  +  b  —  e=2(s  —  c^. 

In  like  manner,       a  —  b  -\-  c  =  2  (^s  —  b^, 
—  a  +  5  +  (?=2(s-a). 


Substituting, 

Show  that 
also 
From 

show  that 
also 


o„in2^_2(«- 

6).2Cs-c) 

-„m  2- 

2  be 

.•.sin|  =  V^- 

sinf=? 

8in|=  ? 

2cos2^  =  l  +  cos^, 


COS 


^      9 

cos-  =  ? 


and 

Also  derive  the  formulae 


~f-' 


^i'^^W^- 


tan:?  =  ? 


tanf=? 


76  PLANE   TRIGONOMETRY. 

63.  Area  of  the  triangle.     In  the  figures  of  Art.  59  the 
area  of  the  triangle  ABC=  A  =  ^pb. 

But     p=csinA.     .*.  A  =  ^hc  sin  A,  (i) 

c  sin  B 


Again,  by  law  of  sines,    h  = 


sin  O 

g^sin^  sin^ 
2  sin  0 

c^sin^sinj^ 


Substituting,  A  = 

Zt  sill  w 

(why?).         (ii) 


2  sin  (^4-^) 

<x         A       A 

Finally,  since  sin^  =  2 sin  —  cos  — ,  we  have  from  (i) 

ju        a 

A       11      o   •    -4.       A      z  ^ls(s  —  a)Cs—b^(s  — c) 

A  =  lbc  '  2 sin—  cos  —  =  bc\-^ ^ — -^^ ^ 

2  2         2  ^  be  '  be 


or  A  =  Vs(s  —  a)(s  — 6)(s  —  <?).        (iii) 

Find  A ;     (1)  Given  a  =  10,  6  =  12,  C  =  45°. 

(2)  Given  a  =  4,  6  =  5,  c  =  6. 

(3)  Given  a  =  2,  B  =  45°,  C  =  60°. 


SOLUTION  OF  TRIANGLES. 

64.   For  the  solution  of  triangles  we  have  the  following 
f ormulee,  which  should  be  carefully  memorized : 

T     sin^  _  sin  B  _  sin  C 
a  h  c 


II.   tan|(^-^)  =  ^^tan|(^  +  B). 


III.    sm  —  =  \^ ^ ^,  or  cos  —  =  ^-^ — ^» 

2       ^  be  2       ^       be 


ortan:|=V^^^H^-o). 
2      ^      s(s-  a) 


IV.   li=lbcs\nA  =  ^^^r^^/^^  =  V8i8-aK8-bKs-c). 
^  2  sm  (A  4-  B) 


SOLUTION  OF   TRIANGLES.  77 

Which  of  the  above  formulae  shall  be  used  in  the  solution 
of  a  given  triangle  must  be  determined  by  examining  the 
parts  known,  as  will  appear  in  Art.  69.  It  is  always  pos- 
sible to  express  each  of  the  unknown  parts  in  terms  of  three 
known  parts. 

In  solving  triangles  such  as  Case  I,  Art.  58,  the  law  of 
sines  applies;  for,  if  the  given  side  is  not  opposite  either 
given  angle,  the  third  angle  of  the  triangle  is  found  from 
the  relation  A  -h  B  -\-  0  =  180°,  and  then  three  of  the  four 

quantities  in  — —  =  ^^ —  being  known,  the  solution  gives 

a  0 

the  fourth. 

In  Case  II  (5)  the  law  of  sines  applies,  but  in  II  (a)  two 

only  of   the   four  quantities   in  ^^2 —  =  EE —  are  known. 

a  0 

Therefore,  we  resort  to  the  formula 

tan  i(^  -B)  =  ^nitanK^  +  B), 

in  which  all  the  factors  of  the  second  member  are  known. 
In  Case  III,  tan  ^  =  >|^~  ^^  ^^  ~  ""^  is  clearly  applicable, 

A  S  yS  —  d) 

and  is  preferred  to  the  formulae  for  sin—  and  cos  —  ;  for, 

first,  it  is  more  accurate  since  tangent  varies  in  magnitude 
from  0  to  00,  while  sine  and  cosine  lie  between  0  and  1. 
(See  Art.  2T,  5.) 

Let  the  student  satisfy  himself  on  this  point  by  finding,  correct  to 
seconds,  the  angle  whose  logarithmic  sine  is  9.99992,  and  whose  loga- 
rithmic tangent  is  1.71668.  Does  the  first  determine  the  angle  ?  Does 
the  second? 

And,  second,  it  is  more  convenient,  since  in  the  complete 
solution  of  the  triangle  by  sin  --  nx  logarithms  must  be  taken 

A  A 

from  the  table,  by  cos  —  seven^  and  by  tan  —  but  four. 

The  right  triangle  may  be  solved  as  a  special  case  by  the 
law  of  sines,  since  sin  (7=1. 


T8 


PLANE   TRIGONOMETRY. 


65.  Ambiguous  case.  In  geometry  it  was  proved  that  a 
triangle  having  two  sides  and  an  angle  opposite  one  of  them 
of  given  magnitude  is  not  always  determined.  The  marks 
of  the  undetermined  or  ambiguous  triangle  are  : 

1.  The  parts  given  are  two  sides  and  an  angle  opposite  one, 

2.  The  given  angle  is  acute. 

3.  The  side  opposite  this  angle  is  less  than  the  other  given 
side. 

When  these  marks  are  aJl  present,  the  number  of  solutions 
must  be  tested  in  one  of  two  ways  : 

((^)  P>om  the  figure  it  is  apparent  that  there  will  be  no 
solution  when  the  side  opposite  is  less  than  the  perpendicular 
p  ;  one  solution  when  side  a  equals  p  ;  and  two  solutions  when 
a  is  greater  than  p. 


M.  b  0    A  b  O  A  b      C  C 

No  Solution,  One  Solution,  Two  Solutions, 

Fig.  35. 

And  since  sin^  =  — ,  it  follows  that  there  will  be  no  solu- 
c 

tion,  one  solution,  two  solutions,  according  as  sin  A  =  — 

<  c 

(5)  A  good  test  is  found  in  solving  by  means  of  loga- 
rithms ;  and  there  will  be  no  solutions,  one  solution,  two  solu- 
tions, according  as  log  sin  O  proves  to  be  impossible,  zero, 
possible,  i.e.  as  log  sin  Q  is  positive,  zero,  or  negative.  This 
results  from  the  fact  that  sine  cannot  be  greater  than  unity, 
whence  log  sine  must  have  a  negative  characteristic,  or  be 
zero. 


66.  In  computations  time  and  accuracy  assume  more  than 
usual  importance.  Time  will  be  saved  by  an  orderly  arrange- 
ment of  the  formulae  for  the  complete  solution,  before  open- 
ing the  book  of  logarithms,  thus  : 


SOLUTION  OF  TRIANGLES.  79 


Given  J.,  B^  a. 

Solve  completely. 

=  180°-(^-h^), 

,      a  sin  -B            « sin  (7     A      1     1   •    /> 
sin  A               sin  J.             ^ 

180° 

log  a=                           log  a  = 

A-\-B  = 

log  sin  jB  =                        log  sin  C  = 

.-.  C  = 

colog  sin  A  =                     colog  sin  A  = 

log  6  =                               log  c  = 

.-.6=                                .-.0  = 

Check: 

loga  = 

l0g(5-&)  = 

log  ft  = 

log(5-c)  = 

log  sin  C  = 

colog  5  = 

colog  2  = 

^                      colog  (5  —  a)  = 

logA  = 

2) 

.•.A  = 

logtan:|  = 

.-.  A  = 

67.  Accuracy  must  be  secured  by  checks  on  the  work  at 
every  step  ;  e.g.  in  adding  columns  of  logarithms,  first  add 
up,  and  then  check  by  adding  down.  Too  much  care  can- 
not be  given  to  verification  in  the  simple  operations  of 
addition,  subtraction,  multiplication,  and  division.  A  final 
check  should  be  made  by  using  other  formulse  involving  the 
parts  in  a  different  way,  as  in  the  check  above.  As  far  as 
possible  the  parts  originally  given  should  be  used  through- 
out in  the  solution,  so  that  an  error  in  computing  one  part 
may  not  affect  later  computations. 

68.  The  formulae  should  always  be  solved  for  the  unknown 
part  before  using^  and  it  should  be  noted  whether  the  solu- 
tion gives  one  value,  or  more  than  one,  for  each  part;  e.g. 
the  same  value  of  sin^  belongs  to  two  supplementary  angles, 
one  or  both  of  which  may  be  possible,  as  in  the  ambiguous 
case. 

"  69.  Write  formulae  for  the  complete  solution  of  the  fol- 
lowing triangles,  showing  whether  you  find  no  solution,  one 
solution,  two  or  more  solutions,  in  each  case,  with  reasons  for 
your  conclusion : 


80 


PLANE   TRIGONOMETRY. 


a 

b 

c 

A 

B 

C 

1. 

81°  26' 28'' 

44°  11' 20" 

54°  22' 12" 

2. 

78.54 

63°  18'  20" 

41°  30' 18" 

3. 

135.82 

26.89 

53°  28' 30" 

4. 

0.75 

0.85 

0.95 

5. 

243 

562 

36°  15'  40" 

6. 

38.75 

25.92 

63°  50'  10" 

7. 

0.058 

78°  15' 

33°  46' 

8. 

2986 

1493 

30° 

9. 

48 

50 

26°  15' 

MODEL  SofctJTIONS. 
1.   Given  a  =  0.785,  b  =  0.85,   c  =  0.633.    Solve  completely. 


tan 


^a/5 


6)0 


c)  ^      B 
— ,  tan  2 


4 


(s  —  a){s  —  c) 


tan 


4 


(s-a)(s-b) 


2  ~  '^     s(s-a)     '         2~^      s(s-b)      '         2       ^      s{s-c) 
Check:  A  +  B  ■}- C  =  180°.     A  =  Vs(s  -  a){s  -  b)(s  -  c). 


a  =  0.735 
b  =  0.85 
c  =  0.633 


2)2.268 

s  =  1.134 

s-a  =  0.349 

s-b  =  0.284 

s  -  c  =  0.501 


log(s-&)=  9.45332 

log  (s  -  c)  =  9.69984 

cologs=  9.94539 

colog  (s  -a)=  0.45717 

2)19.55572 
log  tan  ^^=    9.77786 

^A  =30°  56' 49" 
A  =  61°  53'  38" 


Check:  log(s-a)=  9.54283 

A  =  61°  53'  38"  log  (s  -  &)  =  9.45332 

5  =  72°  46'   4"  colog  s  =  9.94539 

C  =  45°  20'  20"  colog  (s-c)=  0.30016 


180°   0'    2" 


2)19.24170 
logtan^C=    9.62085 

22°  40' 10" 
C  =  45°  20' 20" 


hC 


log(s-a) 

log  (s  -  c) 

colog  s 

colog  (s  —  b) 

log  tan  ^  J5 

IB 

B 

logs 
log  (s  -  a) 
log  (s  -  b) 
log  (s  -  c) 


log  A 

A 


=  9.54283 

=  9.69984 

=  9.94539 

=  0.54668 

2)19.73474 
=    9.86737 
=  36°  23'  2" 
=  72°  46  4" 

=  0.05461 

=  9.54283 

=  9.45332 

=  9.69984 

2)18.75060 
=  9.37530 
=    0.2373 


Solve  :(1)  Given  a  =  30,  &  =  40,  c  =  50. 

(2)  Given  a  =  2159,  b  =  1431.6,  c  =  914.8. 

(3)  Given  a  =  78.54,  b  =  32.56,  c  =  48.9. 


SOLUTION  OF  TRIANGLES. 


81 


2.   Given  A  =  57°  23'  12",  C = 68°  15'  30",  c  =  832.56.    Solve  completely. 


csin  J. 


B 


180 

54°  21'  18". 


sin  C 
(A  +  C) 


b  = 


csinB 


sm  C 
Check :  tan  ^  A 


^  be  sin  A . 
^^(s-bXs-c) 


logc  =  2.92042 

log  sin  A  =  9.92548 

colog  sin  C  =  0.03204 

log  a 


Check . 


2.87794 
a=    754.98 

a=  754.98 
b=  728.38 
c=    832.56 


log  c  =  2.92042 

log  sin  B  =  9.90990 

colog  sin  C  =  0.03204 

log6  =  2.86236 
b=    728.38     A 


s{s  —  a) 

logb=    2.86236 

logc=    2.92042 

log  sin  ^  =    9.92548 

log2A=    5.70826 
=  510811^255405.5 


2)2315.92 
s  =  1157.96 


s-a=    402.98        log(s-6)=  2.63304 

s-  b=    429.58        log (s  -  c)  =  2.51242 

5-c=    325.40  colog  s=  6.93634 

colog  (s- a)  =  7.39471 

2)19.47651 

log  tan  i^  =    9.73826 

^^=28°  41' 38" 

A  =  57°  23'  16" 
Solve : 

(1)  Given  a  =  215.73,  B  =  92°  15',  C  =  28°  14'. 

(2)  Given  b  =  0.827,  A  =  78°  14'  20",  B  =  63°  42'  30". 

(3)  Given  b  =  7.54,  c  =  6.93,  B  =  54°  28'  40". 

3.   Given  a  =  25.384,  c  =  52.925,  5  =  28°  32'  20".    Solve  completely. 
("Why  not  use  the  same  formulae  as  in  Example  1,  or  2?) 


tan 


C-A 


£-:^tan.^+^ 


b  = 


csin^ 


2  c  +  a  2 

180°  -B  =  C-\-A=  151° 27' 40". 


sinC' 

Check:  b  = 


I  ac  sin  B. 
asiiiB 


sin  J 


.'.  ^(C  +  A)=    75°  43' 50". 

c=  52.925  log (c- a)  =  1.43998     .-.  l(C-A)=  54°  7'38" 

a=  25.384         colog  (c  + a)  =8.10619         ^(C+A)=  75°43'50" 

c+a=  7SM9  logtanKC+^)  =  0.59460         ^^^^^^^  C=129°51'28" 

c-a=  27.541  log  tan  i(C-^)  =  0.14077  subtracting,  ^=  21°36'12" 

log  a  =  1.40456 


log  c  =  1.72366 

log  sin  5  =  9.67921 

colog  sin  C= 0.1 1484 

log  &  =  1.51771 
b=  32.939 


Check:  log  a  =  1.40456 

log  sin  5  =  9.67921 

colog  sin  .4  =0.43395 

log  6  =  1.51772 


logc  =  1.72366 
log  sin  5  =  9.67921 

log  2  A =2.70743 
A=511!^=254.965 


82  PLANE  TRIGONOMETRY. 

Solve :    (1)    Given  a  =  0.325,    c  =  0.426,    B  =  48°  50'  10". 

(2)  Given  b  =  4291,     c  =  3194,     A  =  73°  24'  50". 

(3)  Given  b  =  5.38,     c  =  12.45,   A  =  62°  14'  40". 

4.  Ambiguous  eases.  Since  the  required  angle  is  found 
in  terms  of  its  sine,  and  since  sin  a  =  sin  (180°  —  a),  it  fol- 
lows that  there  may  be  two  values  of  a,  one  in  the  first,  and 
the  other  in  the  second  quadrant, ^eir  sum  being  180°.  In 
the  following  examples  the  student  should  note  that  all  the 
marks  of  the  ambiguous  case  are  present.  The  solutions  will 
show  the  treatment  of  the  ambiguous  triangle  having  no 
solution,  one  solution,  two  solutions. 

(a)  Given  5  =  70,  c  =  40,  C=  47°  32'  10''.  Solve.  Why 
ambiguous  ? 

.     p  ^  ^  sin  (7  log  5  =1.84510 

^^^  c     *  logsin  (7=  9.86788 

cologc  =  8.89794 

log  sin^  =  0.11092 

.'.  B  is  impossible,  and  there  is  no  solution.  Why? 
Show  the  same  by  sin  0  >  -• 

(5)  Given  a  =  1.5,  c  =  1.7,  A  =  61°  55'  38".     Solve. 

.    ^^csinA  log  c=  0.23045 

^^^  a     '  logsin^=  9.94564 

colog«=  9.82391 

logsin  (7=  0.00000 
a=90° 


and   there  is  one  solution.     Why  ?     Show   the  same    by 

sin  A  - 

work. 


sin  J.  =  -.      Solve  for  the  remaining  parts  and  check  the 

0 


SOLUTION  OF   TRIANGLES.  83 

(c)  Given  a  =  0.235,  b  =  0.189,  B  =  36°  28'  20^'.     Solve. 

.      .      a  sin  ^  b  sin  O 

sin  A  =  — = ?  e  = —-. 

0  sin^ 

log  a  =9. 37107  log  6  =  9. 27646  9. 27646 

log  sin  ^  =  9. 77411  log  sin  0=9. 99772  or  9. 28774 

colog  b  =  0.72354  colog  sin  B  =  0.22589  0. 22589 

log  sin  ^  =  9.86872  log  c  =  9.50007  or  8.79009 

A  =  47°  39'  25''  c  =  0.31628  or  0.06167 

or  132°  20'  35". 

...  (7  =95°  52'  15"  or  11°  11'  5". 

Solve  for  A,  and  check.     Show  the  same  by  sin  B  <  — 
Solve  : 

(1)  Given  6  =  216.4,  c=  593.2,  B=  98°  15'. 

(2)  Given  a  =  22,  6  =  75,  5  =  32°  20'. 

(3)  Given  a  =  0.353,  c=  0.295,  A  =  46°  15'  20". 

(4)  Given  a  =  293.445,  b  =  450,  A  =  40°  42'. 

(5)  Given  b  =  531.03,  c=  629.20,  ^=34°  28'  16". 


Solve  completely,  given : 

a 

h 

c 

A 

B                  C 

L           50 

60 

78°  27' 47" 

2. 

10 

11 

93°  35' 

3.             4 

5 

6 

4. 

10 

109°  28' 16" 

38°  56' 54" 

5.           40 

51 

49°  28' 32" 

6.     352.25 

513.27 

482.68 

7.       0.573 

0.394 

112°   4' 

8.   107.087 

56°  15' 

48°  35' 

9. 

V2 

117° 

45° 

10.     197.63 

246.35 

34°  27' 

11.        4090 

3850 

3811 

12.        3795 

73°  15' 15"    42°  18' 30" 

13. 

234.7 

185.4 

84°  36' 

14. 

26.234 

22.6925 

49°   8' 24" 

15.         273 

136 

72°  25' 13" 

84  PLANE   TRIGONOMETRY. 

APPLICATIONS. 

70.  Measurements  of  heights  and  distances  often  lead  to 
the  solution  of  oblique  triangles.  With  this  exception,  the 
methods  of  Chapter  V  apply,  as  will  be  illustrated  in  the 
following  problems. 

The  hearing  of  a  line  is  the  angle  it  makes  with  a  north 
and  south  line,  as  determined  by  the^magnetic  needle  of  the 
mariner's  compass.  If  the  bearing  does  not  correspond  to 
any  of  the  points  of  the  compass,  it  is  usual  to  express  it 
thus:  N.  40°  W.,  meaning  that  the  line  bears  from  N.  40° 
toward  W. 

EXAMPLES. 

1.  When  the  altitude  of  the  sun  is  48°,  a  pole  standing  on  a  slope 
inclined  to  the  horizon  at  an  angle  of  15°  casts  a  shadow  directly  down 
the  slope  44.3  ft.     How  high  is  the  pole? 

2.  A  tree  standing  on  a  mountain  side  rising  at  an  angle  of  18°  30' 
breaks  32  ft.  from  the  foot.  The  top  strikes  down  the  slope  of  the  moun- 
tain 28  ft.  from  the  foot  of  the  tree.     Find  the  height  of  the  tree. 

3.  From  one  corner  of  a  triangular  lot  the  other  corners  are  found  to 
be  120  ft.  E.  by  N.,  and  150  ft.  S.  by  W.  Find  the  area  of  the  lot,  and 
the  length  of  the  fence  required  to  enclose  it. 

4.  A  surveyor  observed  two  inaccessible  headlands,  A  and  B.  A  was 
W.  by  N.  and  B,  N.E.  He  went  20  miles  N.,  when  they  were  S.W.  and 
S.  by  E.     How  far  was  A  from  B  ? 

5.  The  bearings  of  two  objects  from  a  ship  were  N.  by  W.  and  N.E. 
by  N.  After  sailing  E.  11  miles,  they  were  in  the  same  line  W.N.W. 
Find  the  distance  between  them. 

6.  From  the  top  and  bottom  of  a  vertical  column  the  elevation  angles 
of  the  summit  of  a  tower  225  ft.  high  and  standing  on  the  same  hori- 
zontal plane  are  45°  and  55°.     Find  the  height  of  the  column. 

7.  An  observer  in  a  balloon  1  mile  high  observes  the  depression  angle 
of  an  object  on  the  ground  to  be  35°  20'.  After  ascending  vertically  and 
uniformly  for  10  mins.,  he  observes  the  depression  angle  of  the  same  object 
to  be  55°  40'.     Find  the  rate  of  ascent  of  the  balloon  in  miles  per  hour. 

8.  A  statue  10  ft.  high  standing  on  a  column  subtends,  at  a  point 
100  ft.  from  the  base  of  the  column  and  in  the  same  horizontal  plane,  the 
same  angle  as  that  subtended  by  a  man  6  ft.  high,  standing  at  the  foot 
of  the  column.     Find  the  height  of  the  column. 

9.  From  a  balloon  at  an  elevation  of  4  miles  the  dip  of  the  horizon 
is  2°  33'  40".     Required  the  earth's  radius. 


TRIANGLES  —  APPLICATIONS.  85 

10.  Two  ships  sail  from  Boston,  one  S.E.  50  miles,  the  other  N.E.  by 
E.  60  miles.  Find  the  bearing  and  distance  of  the  second  ship  from  the 
first. 

11.  The  sides  of  a  valley  are  two  parallel  ridges  sloping  at  an  angle  of 
30°.  A  man  walks  200  yds.  up  one  slope  and  observes  the  angle  of  eleva- 
tion of  the  other  ridge  to  be  15°.  Show  that  the  height  of  the  observed 
ridge  is  273.2  yds. 

12.  To  determine  the  height  of  a  mountain,  a  north  and  south  base 
line  1000  yds.  long  is  measured ;  from  one  end  of  the  base  line  the  sum- 
mit bears  E.  10°  N.,  and  is  at  an  altitude  of  13°  14'.  From  the  other  end 
it  bears  E.  46°  30'  N.     Find  the  height  of  the  mountain. 

13.  The  shadow  of  a  cloud  at  noon  is  cast  on  a  spot  1600  ft.  due  west 
of  an  observer.  At  the  same  instant  he  finds  that  the  cloud  is  at  an  ele- 
vation of  23°  in  a  direction  W.  14°  S.  Find  the  height  of  the  cloud  and 
the  altitude  of  the  sun. 

14.  From  the  base  of  a  mountain  the  elevation  of  its  summit  is  54°  20'. 
From  a  point  3000  ft.  toward  the  summit  up  a  plane  rising  at  an  angle 
of  25°  30'  the  elevation  angle  is  68°  42'.    Find  the  height  of  the  mountain. 

15.  From  two  observations  on  the  same 

meridian,   and  92°  14'   apart,   the   zenith 

angles  of  the  moon   are   observed  to   be 

44°  54' 21"    and   48°  42' 57".      Calling   the 

earth's  radius  3956.2  miles,  find  the   dis-      ,        ,,    ,  ^ 

,  ,     ,,  {        iC.\/\Z= Zenith  angle 

tance  to  the  moon.  '  ^  \^    ^  v 

16.  The  distances  from  a  point  to  three 
objects  are  1130,  1850, 1456,  and  the  angles 

subtended  by  the  distances  between  the  three  objects  are  respectively 
102°  10',  142°,  and  115°  50'.    Find  the  distances  between  the  three  objects. 

17.  From  a  ship  A  running  N.E.  6  mi.  an  hour  direct  to  a  port  dis- 
tant 35  miles,  another  ship  B  is  seen  steering  toward  the  same  port,  its 
bearing  from  A  being  E.S.E.,  and  distance  12  miles.  After  keeping  on 
their  courses  1^  hrs.,  B  is  seen  to  bear  from  A  due  E.  Find  B's  course 
and  rate  of  sailing. 

18.  From  the  mast  of  a  ship  64  ft.  high  the  light  of  a  lighthouse  is 
just  visible  when  30  miles  distant.  Find  the  height  of  the  lighthouse, 
the  earth's  radius  being  3956.2  miles. 

19.  From  a  ship  two  lighthouses  are  observed  due  N.E.  After  sailing 
20  miles  E.  by  S.,  the  lighthouses  bear  N.N.W.  and  N.  by  E.  Find  the 
distance  between  the  lighthouses. 

20.  A  lighthouse  is  seen  N.  20°  E.  from  a  vessel  sailing  S.  25°  E.  A 
mile  further  on  it  appears  due  N.  Determine  its  distance  at  the  last 
observation. 


EXAMPLES  FOR  ^VIEW. 

In  connection  with  each  problem  the  student  should  review 
all  principles  involved.  The  following  list  of  problems  will  then 
furnish  a  thorough  review  of  the  book.  In  solving  equations, 
find  all  values  of  the  unknown  angle  less  than  360°  that  satisfy 
the  equation. 

1.  If  tan  «  =  },  tan  /?  =  i,  show  that  tan  (/3  -  2  a)  =  -j^. 

2.  Prove  tan  a  +  cot  a  =  2  esc  2  a. 

A  A  A       A 

3.  From  the  identities  sin^ — |-  cos^—  =  1,  and  2  sin  —  cos—  =  sin  A. 

2  2'  22 

prove  2  sin  —  =  ±  Vl  +  siii  A  ±Vl  —  sin  A, 

and  2  cos  —  =  ±  Vl  +  sin  A  T  Vl  -  sin^. 

4.  Remove  the  ambiguous  signs  in  Ex.  3  when  A  is  in  turn  an  angle 
of  each  quadrant. 

5.  A  wall  20  feet  high  bears  S.  59°  5'  E. ;  find  the  width  of  its  shadow 
on  a  horizontal  plane  when  the  sun  is  due  S.  and  at  an  altitude  of  60°. 

6.  Solve  sin  a:  +  sin  2  a:  +  sin  3  a;  =  1  +  cos  x  +  cos  2  x. 

7.  Prove  tan-i  i  +  tan-i  i  =  ^. 

8.  If  ^  =  60°,  B  =  45°,  C  =  30°,  evaluate 

tan  A  +  tan  B  +  tan  C 


tan  A  tan  B  -!-  tan  B  tan  C  +  tan  C  tan  A 

9    Prove  ^Q^  (^^  +  ^)  <^os  C  _  1  —  tan  A  tan  B 
cos  (A  +  C)  cos  B     1  —  tan  A  tan  C 

10.  Solve  completely  the  triangle  whose  known  parts  are  b  =  2.35, 
c  =  1.96,  C  =  38°  4:5' A. 

11.  Find  the  functions  of  18°,  36°,  54°,  72°. 

Let  x  =  18°.    Then  2a;=36°,  3x  =  54°,  and  2x  +  3a  =  90°. 

P 

12.  If  cot  a  =  -,  find  the  value  of 

sin  a  +  cos  a  +  tan  a  +  cot  a  +  sec  a  +  esc  a. 
86 


EXAMPLES  FOR  REVIEW.  87 

T «     -r,  sin  3  « sin  2  ^  —  sin  3  i8  sin  2  a      -,    ,  ^  o 

13.  Prove  — ^— ■, — ^ : — -^-^ =  1  +  4  cos  a  cos  B. 

sin  2  a  sin  p  —  sin  2  /j  sin  a  ' 

14.  From  a  ship  sailing  due  N.,  two  lighthouses  bear  N.E.  and 
N.N.E.,  respectively;  after  sailing  20  miles  they  are  observed  to  bear 
due  E.     Find  the  distance  between  the  lighthouses. 

15.  Solve  1  —  2  sin  a:  =  sin  3  x, 

16.  Prove  sin-i\'— ^  =  tan-i-\p. 

^a  +  b  ^b 

17.  If  cos  ^  —  sin  ^  =  \/2  sin  6,  then  cos  ^  +  sin  ^  =  V2  cos  9, 

18.  Solve  completely  the  triangle  ABC,  given  a  =  0.256,  b  =  0.387, 
C  =  102°  20'.5. 

2  cos  2  cc  -  1 


19.   Prove  tan  (30°  +  a)  tan  (30°-  a)  = 


2  cos  2  a  +  1 

20.  Solve  tan  (45°  -  0)  +  tan  (45°  +  ^)  =  4. 

21.  Prove  sin^  a  cos^  /8  -  cos^  a  sin^  ft  =  sin^  a  -  sin^  ^. 

22.  Prove  cos^  a  cos^  /3  -  sin^  a  sin^  ^  =  cos^ «  -  sin^  p.. 

23.  A  man  standing  due  S.  of  a  water  tower  150  feet  high  finds  its 
elevation  to  be  72°  30' ;  he  walks  due  W.  to  A  street,  where  the  elevation 
is  44°  50' ;  proceeding  in  the  same  direction  one  block  to  B  street,  he  finds 
the  elevation  to  be  22°  30'.  What  is  the  length  of  the  block  between  A 
and  B  streets. 

24.  Prove  tan-i  -  4-  tan-i  -  +  tan-i  i  +  tan-i  i  =  -• 

3  5  7  8      4 

25.  If  P  =  60°,  Q  =  45°,  R  =  30°,  evaluate 

sin  P  cos  Q  +  tan  P  cos  Q 
sin  P  cos  P  +  cot  P  cot  R 

26.  If  cos  (90°  +  «)  =  -!,  evaluate  3  cos  2  a  +  4  sin  2  a. 

27.  If  sin  B  +  sin  C  =  m,  cos  J5  +  cos  C  =  n,  show  that  tan  — ^ —  =  — . 

2  n 

28.  Show  that  sin  2  )3  can  never  be  greater  than  2  sin  )8. 

29.  Prove  sin-^  |  +  sin-^  ^  =  tan-^  f  f . 

30.  Solve  cot-ix  +  sin-i- V5  =  ^• 

o  4 

31.  Solve  sin-^x  +  sin-i(l  —  x)=  cos~^ar. 

32.  A  man  standing  between  two  towers,  200  feet  from  the  base  of 
the  higher,  which  is  90  feet  high,  observes  their  altitudes  to  be  the  same ; 
70  feet  nearer  the  shorter  tower  he  finds  the  altitude  of  one  is  twice  that 
of  the  other.  Find  the  height  of  the  shorter  tower,  and  his  original 
distance  from  it. 


88  PLANE   TRIGONOMETRY. 

33.  Solve  cos  3  /3  +  8  cos^  p  =  0. 

34.  Solve  cot  m  —  tan  (180  +  m)  =  sin  m  +  sin  (90"  —  m), 

35.  Solve  ljzi!:Bi  =  2  cos  2 1. 

1  +  tan « 

36.  Prove  cot^  +  cot  5  =^^-(A±M. 

sm  A  sm  B 

37.  Prove  cot  P  -  cot  Q  =  -  ^^"^^7  ^^- 

sin  P  sm  Q 

38.  In  the  triangle  ABC  prove 

a  =  6  sin  C  +  c  sin  5, 
6  =  c  sin  J.  +  a  sin  C, 
c  =  a  sin  .B  +  5  sin  ^4. 

39.  Solve  completely  the  triangle,  given 

a  =  927.56,  b  =  648.25,  c  =  738.42. 

40.  Prove  cos^  a  -  sin  (30°  +  a)  sin  (30°  -  «)  =  f . 

-,     -D  X      o     J.  cos  2  a;  —  cos 4  a: 

41.  Prove  tan  3  x  tan  a;  = — 

cos  2  a;  +  cos  4  x 

42.  Simplify  cos  (270°  +  «)  +  sin  (180°  +  a)+  cos  (90°  +  a). 

43.  Simplify  tan  (270°  -$)-  tan  (90°  +  6)+  tan  (270°  +  0). 

44.  Solve  cos  3  <^  —  cos  2  <^  +  cos  ^  =  0. 

45.  Solve  cos  ^  +  cos  3  ^  +  cos  5  ^  +  cos  7  ^  =  0. 

46.  The  topmast  of  a  yacht  from  a  point  on  the  deck  subtends  the 
same  angle  a,  that  the  part  below  it  does.  Show  that  if  the  topmast  be 
a  feet  high,  the  length  of  the  part  below  it  is  a  cos  2  a. 

47.  A  horizontal  line  AB  is  measured  400  yards  long.  From  a  point 
in  A  B  Si  balloon  ascends  vertically  till  its  elevation  angles  at  A  and  B 
are  64°  15'  and  48°  20',  respectively.     Find  the  height  of  the  balloon. 

sin  a 


48.  If  cos  d)  =  n  sin  a,  and  cot<^=       '*   prove  cos  B= 

tan^  Vl  +  n2cos2a 

49.  Find  cos  3  a,  when  tan  2  a  =  —  f . 

50.  Solve    completely  the  triangle,  given  a  =  0.296,  B  =  28°  47'.3, 
C  =  84°  25'. 

51.  Evaluate  sin  300°  +  cos  240°  +  tan  225^ 

52.  Evaluate  sec  IS  -  esc  ^  +  tan  1^. 

O  O  O 


EXAMPLES  FOR  REVIEW.  89 

53.  If  tan^  =  ^^^"^^^V-^"^^^^^Y 

cos  «  COS  y  —  cos  /8  sin  y 

and  tan  <t>  =  sin  a  sin  y  -  sin  ^  cos  y^ 

cos  «  sin  y  —  cos  ^  cos  y 
show  that  tan(^  +  <^)  =  tan(a  +  (3). 

54.  If  tan  466°  15'  38"  =  -  ^^,  find  the  sine  and  cosine  of  233"  7'  49". 
55    Prove  ^^^  ^  ~  ^^^  ^      sec  a  —  tan  a 


sec  a  +  tan  a     esc  a  +  cot  a 
56. 


Prove  cos((.-3^)-cos(3«-^)  ^  ^  sin(a  -  fl). 
sin  2 «  +  sin  2  ^  ^        '^^ 

57.  Prove  sin  80°  =  sin  40°  +  sin  20°. 

58.  Prove  cos  20°  =  cos  40°  +  cos  80°. 

59.  Prove  4  tan-i  -  -  tan"!  —  =  S 

5  239      4 

60.  From  the  deck  of  a  ship  a  rock  bears  N.N.W.  After  the  ship 
has  sailed  10  miles  E.N.E.,  the  rock  bears  due  W.  Find  its  distance 
from  the  ship  at  each  observation. 

61.  Find  the  length  of  an  arc  of  80°  in  a  circle  of  4  feet  radius. 

62.  Given  tan  6  =  ^,  tan  <f}  =  -^^,  evaluate  sin(^  +  <^)  +  cos(^  —  <^). 

63.  If  tan  ^  =  2  tan  <^,  show  that  sin(^  +  <^)  =  3  sin(^  -  <^). 

64.  Prove  cos(a  +  j8)cos(a-^)  +  sin(a  +  /8)sin(a-fi)=i^I*^^. 

^       '^  '^       l+tan2^ 

65.  Solve  4  cos  2  ^  +  3  cos  ^  =  1. 

66.  Solve  3  sin  a  =  2  sin  (60°  -  a). 

67.  Prove  (sin  a  -  esc  «)2  —  (tan  a  -  cot  a)  ^  +  (cos  a  —  sec  «) 2=  1. 

68.  Prove  2(sin^  a  +  cos^  a)  +  1  =  3(sin^  a  +  cos*  a). 

69.  Prove  esc  2  y8  +  cot  4  /?  =  cot  )8  -  esc  4  p. 

70.  If  tan  »  =  — ,  cos  2  g  =  — ,  then  esc  ^^-^  =  SVlS. 

^      12  ^      625  2 

71.  Solve  completely  the  triangle,  given 

a  =  0.0654,   6  =  0.092,   5  =  38°40'.4. 

72.  Solve  completely  the  triangle,  given 

&  =  10,   c  =  26,   J5  =  22°37'. 

73.  A  railway  train  is  travelling  along  a  curve  of  |  mile  radius  at  the 
rate  of  25  miles  per  hour.  Through  what  angle  (in  circular  measure) 
will  it  turn  in  half  a  minute  ? 


90  PLANE  TRIGONOMETRY. 

74.  Express  the  following  angles  in  circular  measure : 

63°,    4°  30',     6°  12' 36". 

75.  Express  the  following  angles  in  sexagesimal  measure : 

6       8 '      64 ' 

76.  Ji  A,  B,  C  are  angles  of  a  triangle,  prove 

ABC 

cos  A  +  cos  -B  +  cos  C  =  1  +  4  sin  —  sin  —  sin  -• 

77.  Prove  sin  2  x  +  sin  2  y  +  sin  2  2;  =  4  sin  a;  sin  y  sin  z,  when  a?,  y,  z 
are  the  angles  of  a  triangle. 

78.  Prove  sec  a  =  1  +  tan  a  tan  -• 

79.  Prove  sin^  («  +  j8)  -  sin^  (a  _  ^)  =  sin  2  a  sin  2  j8. 

80.  Prove  cos^  (a  +  /3)  -  sin^  (^a  -  f3)=  cos  2  cc  cos  2  ^. 

81.  Prove  sinl9  ;>  +  sin  17  p  ^  2  cos  9;,. 

sm  10  p  +  sm  8j9 

82.  Consider  with  reference  to  their  ambiguity  the  triangles  whose 
known  parts  are : 

(a)  a  =  2743,  b  =  6452,     B  =  43°  15' ; 

(b)  a  =  0.3854,  c  =  0.2942,    C=:38°20^ 

(c)  &=    5,  c  =  53,  5  =  15°22'; 

(d)  a  =  20,  b  =  90,  A=  63°  28'.5. 

83.  From  a  ship  at  sea  a  lighthouse  is  observed  to  bear  S.E.  After 
the  ship  sailed  N.E.  6  miles  the  bearing  of  the  lighthouse  is  S.  27°  30'  E. 
Find  the  distance  of  the  lighthouse  at  each  time  of  observation. 

84.  Prove  sin  (^  +  3  <^)  +  sin  (3  ^  +  <^)  ^  3  cos  (0  +  <^). 

sin  2  d  +  sin  2  <^  v    ^  v'y 

85.  Prove  cos  15°  -  sin  15°  =  — • 

V2 

86.  Show  that  cos  (a  +  )8)  cos  (a  —  /3)=  cos^  a  -  sin^  p 

=  cos2)8  — sin^a. 

87.  Show  that  tan  (a  +  45°)  tan  (a  -  45°)  =  ^sin^ot-l 

^  ^        ^  ^     2cos2a-l 

88.  Solve  sin  (x  +  y)  sin  (x  —  y)=  ^,    cos  (x  +  y)  cos  (x  —  y)  =  0. 

89.  Prove  l  +  sin«-cos« ^  ^^^  a 

1  +  sin  a  +  cos  a  2 


EXAMPLES  FOR  REVIEW.  91 

90.  Prove  tan  2  ^  +  sec  2  ^  =  cos  0  + sin  6 

cos  ^  —  sin  ^ 

91.  If  tan  <}>=-,  then  a  cos  2^  +  &sin2d>  =  a, 

a 

92.  Prove  sin-i-i  +  cot-i3  =  ^' 

93.  Solve  cos  A  +  cos  7  A  =  cos  4  A. 

94.  Two  sides  of  a  triangle,  including  an  acute  angle,  are  5  and  7, 
the  area  is  14 ;  find  the  other  side. 

95.  Show  that  3cos3g-2cose-cos5g ^  ^^^ ^ ^ 

sm  5  ^  —  3  sm  3  ^  +  4  sin  6 

96.  A  regular  pyramid  stands  on  a  square  base  one  side  of  which  is 
173.6  feet.  This  side  makes  an  angle  of  67°  with  one  edge.  What  is 
the  height  of  the  pyramid  ? 

97.  From  points  directly  opposite  on  the  banks  of  a  river  500  yards 
wide  the  mast  of  a  ship  lying  between  them  is  observed  to  be  at  an  eleva- 
tion of  10°  28'.4  and  12°  14'.5,  respectively.     Find  the  height  of  the  mast. 

98.  Show  that  (sin  60°  -  sin  45°)  (cos  30°  +  cos  45°)  =  sin2  30°. 

99.  Find  x  if  sin-i  x  +  sin-i  ^  =  ^. 

2     4 

100.   Trace  the  changes  in  sign  and  value  of  sin  a  +  cos  a  as   a 
changes  from  0°  to  360°. 


FIVE-PLACE 


LOGARITHMIC  AND  TRIGONOMETRIC 


TABLES 


ADAPTED  FROM  GAUSS'S  TABLES  i 

i 

BY  j 

ELMER   A.    LYMAN  J 

MICHIGAN   STATE  NORMAL  COLLEGE  • 

AND  ^ 

EDWIN   C.    GODDARD  | 

UNIVERSITY    OF  MICHIGAN  i 


>J«io 


ALLYN    AND    BACON 
Boston  antJ  Chicago 


V/^ 


COPYRIGHT,  189  9,  BY 
ELMER  A.  LYMAN  and 
EDWIN   C.    GODDARD. 


Nortoooti  iPwaa 

J.  S.  Cashing  &  Co.  —  Berwick  &  Smith 
Norwood  Mass.  U.S.A. 


TABLE  I. 

THE  COMMON  LOGARITHMS  OF  NUMBERS 
FROM  1   TO  10009. 


N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8   9 

P.P. 

100 

00000 

043 

087 

130 

173 

217 

260 

303 

346  389 

lOI 

432 

475 

518 

561 

604 

647 

689 

732 

775  817 

44  43  42 

I02 

860 

903 

945 

988 

*03o 

*072 

*ii5  *i57  *i99  *242 

I 
2 

103 

01  284 

326 

368 

410 

452 

494 

536 

tl 

620  662 

4/4  4/3  4/2 
8,8  8,6  8,4 
13,2  12,9  12,6 
17,6  17,2  16,8 
22,0  21,5  21,0 
26,4  25,8  25,2 
30,8  30,1  29,4 
35/2  34,4  33,6 
39/6  38,7  37,8 

104 

703 

74S 

787 

828 

870 

912 

953 

^036  ^078 

3 

4 

105 

02  119 

160 

202 

243 

284 

325 

366 

407 

449  490 

106 

531 

572 

612 

653 

694 

735 

776 

816 

857  898 

107 

938 

979  *oi9 

*o6o 

*ioo 

*i4i 

*i8i 

*222 

^262  ^302 

7 
8 

108 

03342 

383 

423 

463 

503 

543 

583 

623 

663  703 

109 

743 

782 

822 

862 

902 

941 

981 

*02I 

*o6o  *ioo 

9 

110 

04139 

179 

218 

258 

297 

336 

376 

415 

454  493 

III 

532 

571 

610 

650 

689 

727 

766 

805 

844  883 

41   40  39 

112 

922 

961 

999 

*038 

*077 

*ii5  *i54 

*i92  *23i  ^269 

113 

05308 

346 

385 

423 

461 

500 

538 

576 

614  652 

I 

4/1  4/0  3/9 
8,2  8,0  7,8 

12.3  12,0  11,7 

16.4  16,0  15,6 

20.5  20,0  19,5 
24/6  24,0  23,4 

28.7  28,0  27,3 

32.8  32,0  31,2 

36.9  36,0  35,1 

114 

690 

729 

767 

805 

843 

881 

918 

956 

'994  *032 

2 
3 
4 
5 
6 

115 

06  070 

108 

145 

183 

221 

258 

296 

333 

371  408 

116 

446 

483 

521 

558 

595 

633 

670 

707 

744  781 

117 

819 

856 

893 

930 

967 

*oo4  *04i 

*078 

*ii5  *i5i 

7 
8 

118 

07  188 

225 

262 

298 

335 

372 

408 

445 

482  518 

119 

555 

591 

628 

664 

700 

737 

773 

809 

846  882 

9 

120 

918 

954 

990  *027 

*o63 

*099 

*i35  *i7i  *207  *243 

1 

121 

08279 

314 

350 

386 

422 

458 

493 

529 

565  600 

38  37  36  1 

122 

636 

672 

707 

743 

778 

814 

849 

884 

920  955 

3/8  3/7  3/6 
7/6  7/4  7/2 
11,4  II, I  10,8 
15,2  14/8  14/4 
19,0  18,5  18,0 

123 

991 

*026 

*o6i 

^096  ^132 

*i67 

*202  *237 

*272  *307 

2 

124 

09342 

377 

412 

447 

482 

517 

552 

587 

621  656 

4 

125 

691 

726 

760 

795 

830 

864 

899 

934 

968  *oo3 

126 

10037 

072 

106 

140 

175 

209 

243 

278 

312  346 

5 

22,8  22,2  21,6 

127 

380 

415 

449 

483 

517 

551 

585 

619 

653  687 

7 

26,6  25,9  25,2 

128 

721 

755 

789 

823 

857 

890 

924 

958 

992  *025 

8 

30,4  29,6  28,8 

129 

II  059 

093 

126 

160 

193 

227 

261 

294 

327   361 

9 

34/2  33,3  32,4 

130 

394 

428 

461 

494 

528 

561 

594 

628 

661   694 

1 

131 

727 

760 

793 

826 

860 

893 

926 

959 

992  ^024 

35  34  33  1 

132 

12  057 

090 

123 

156 

189 

222 

254 

287 

320   352 

133 

38S 

418 

450 

f3 

516 

548 

581 

613 

646   678 

I 
2 
3 

4 

3/5  3/4  3/3 
7,0  6,8  6,6 
10,5  10,2  '9,9 
14,0  13,6  13,2 
17/5  I7/0  16,5 
21,0  20,4  19,8 
24/5  23,8  23,1 
28,0  27,2  26,4 
31/5  30/6  29,7 

134 

710 

743 

775 

808 

840 

872 

905 

937 

969  jifOOI 

135 

13033 

066 

098 

130 

162 

194 

226 

258 

290  322 

136 

354 

386 

418 

450 

481 

513 

545 

577 

609  640 

137 

672 

704 

735 

767 

799 

830 

862 

893 

925  956 

I 

X38 

988 

*oi9  *o5i 

*o82 

*ii4 

*I45 

*I76 

*208 

*239  *270 

139 

14  301 

333 

364 

395 

426 

457 

489 

520 

551  582 

9 

140 

613 

644 

675 

706 

737 

768 

799 

829 

860  891 

141 

922 

953 

983  *oi4  *045 

*076 

*io6 

*I37 

^168  ^198 

32  31   30 

143 

•15  229 

259 

290 

320 

351 

381 

412 

442 

473  503 

143 

534 

564 

625 

655 

685 

715 

746 

776  806 

I 

6,4  6,2  6,0 
9/6  9/3  9,0 
12,8  12,4  12,0 
16,0  15,5  15,0 
19,2  18,6  18,0 
22,4  21,7  21,0 
25,6  24,8  24,0 
28,8  27,9  27,0 

144 

145 

836 

866 

897 

927 

957 

987 

*oi7  *047  *077  *I07 

2 
3 
4 

i 

16  137 

167 

197 

227 

256 

286 

316 

346 

376  406 

146 

43S 

465 

495 

524 

554 

584 

613 

643 

673  702 

147 

732 

761 

791 

820 

850 

879 

909 

938 

967  997 

7 

8 

148 

17026 

056 

085 

114 

143 

173 

202. 

231 

260  289 

149 

319 

348 

377 

406 

435 

464 

493 

522 

551  580 

9 

150 

609 

638 

667 

696 

725 

754 

782 

811 

840  869 

N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8   9 

P.P. 

N. 

L.  0 

1 

2   3 

4 

5 

6 

7   8   9 

»   1 

150 

17609 

638 

667  696 

725 

754 

782 

811  840  869 

151 

898 

926 

955  984 

*oi3 

*04i 

*070  *099  *I27  »i56 

29 

28 

152 

18  184 

213 

241  270 

298 

327 

35S 

384  412  441 

2,8 

5/6 

8,4 

11,2 

14,0 

16,8 

153 

469 

498 

526  554 

583 

611 

639 

667  696  724 

I 

2 
3 
4 

5,8 

8,7 

11,6 

14,5 
17,4 
20.3 

154 

752 

780 

808  837 

865 

893 

921 

949  977  *oo5 

155 

19033 

061 

089  117 

14$ 

173 

201 

229  257  285 

156 

312 

340 

368  396 

424 

45^ 

479 

507  535  562. 

157 

590 

618 

645  673 

700 

728 

756 

783  811  838 

7 
8 

19/6 
22,4 
25,2 

158 

866 

893 

921  948 

976 

*oo3  *030  ^058  *o85  *ii2 

159 

20  140 

167 

194  222 

249 

276 

303 

330  358  385 

9 

160 

412 

439 

466  493 

520 

548 

575 

602  629  656 

161 

683 

710 

737  763 

790 

817 

844 

871  898  925 

27 

26 

162 

952 

978 

*ooS  *032  *o59 

*o85 

*II2 

*I39  *i65  »I92 

2,6 

163 

21  219 

245 

272  299 

325 

352 

378 

405  431  458 

■'■ 

^,/ 

164 

484 

511 

537  564 

590 

617 

643 

669  696  722 

2 
3 
4 
5 
6 

5,4 

8,1 

10,8 

13,5 
16,2 

5,2 

7,8 
10,4 
13,0 
i^  6 

165 

748 

775 

801  827 

854 

880 

906 

932  958  985 

i6b 

22  on 

037 

063  089 

141 

167 

194  220  246 

167 

272 

298 

324  350 

376 

401 

427 

453  479  505 

7 
8 

18^9 

21  6 

i8;2 

168 

531 

557 

583  608 

634 

660 

686 

712  737  763 

20' 8 

169 
170 

789 

814 

840  866 

891 

.  917 

943 

968  994  *oi9 

9 

24^3 

23U 

23045 

070 

096  121 

147 

172 

198 

223  249  274 

171 

300 

325 

350  r  376 

401 

426 

452 

477  502  528 

25 

172 

553 

578 

603  629 

654 

679 

704 

729  754  779 

T 

2 

5 

173 

805 

830 

855  880 

905 

930 

955 

980  j(t005  ^030 

2 

c 

174 

24055 

080 

105  130 

155 

180 

204 

229  254  279 

3 
4 

10,0    1 

175 

304 

329 

353  378 

403 

428 

452 

477  502  527 

12 

5 

176 

551 

601  625 

650 

674 

699 

724  748  773 

5 

15 
17 
20 

0 

177 

797 

822 

846  871 

89s 

920 

944 

969  993  *oi8 

7 

5 

178 

25042 

066 

091  115 

139 

164 

188 

212  237  261 

8 

0 

179 

285 

310 

334  358 

382 

406 

431 

455  479  503 

9 

22 

5 

180 

527 

551 

575  600 

624 

648 

672 

696  720  744 

181 

768 

792 

816  840 

864 

888 

912 

935  959  983 

24 

23 

182 

26007 

031 

055  079 

102 

126 

150 

174  198  221 

I 

2,3 
4,6 
6,9 
9,2 
11,5 
13,8 
16  I 

183 

245 

269 

293  316 

340 

364 

387 

411  435  458 

2,4 

4,8 

9,6 
12,0 

184 

482 

505 

529  553 

576 

600 

623 

647  670  694 

2 
3 

4 
5 
6 

185 

717 

741 

764  788 

811 

834 

858 

881  905  928 

186 

951 

975 

998  *02i  *045 

*o68 

*09i 

*II4  #138  *i6i 

I4!4 
16,8 

187 

27184 

207 

231  254 

277 

300 

323 

346  370  393 

7 
8 

188 

416 

439 

462  485 

508 

531 

554 

577  600  623 

I9!2 
21,6 

i8;4 

20/7 

189 

646 

669 

692  715 

738 

761 

784 

807  830  852 

9 

190 

875 

898 

921  944 

967 

989  *oi2  *o35  ^058  »o8i 

191 

28  103 

126 

149  171 

194 

217 

240 

262  285  307 

22 

21 

192 

■330 

353 

37o  398 

421 

443 

466 

48S  511  533 

193 

556 

578 

601  623 

646 

668 

691 

713  735  758 

I 

2/2 

2/1 

194 

780 

803 

825  847 

870 

892 

914 

937  959  981 

2 

3 
4 
5 
6 

8,8 
11,0 

8/4 
12,6 

195 

29003 

026 

048  070 

092 

"5 

137 

159  181  203 

196 

226 

248 

270  292 

314 

336 

358 

380  403  425 

13^2 

17,6 
19/8 

197 

447 

469 

491  513 

535 

557»  579 

601  623  645 

I 

14^7 
168 

198 

667 

688 

710  732 

754 

776 

798 

820  842  863 

199 
200 

885 

907 

929  951 

973 

994 

*oi6 

^038  *o6o  *o8i 

9 

i8;9 

30103 

125 

146  168 

190 

211 

233 

255  276  298 

N. 

L.  0 

1 

2   3 

4 

5 

6 

7   8   9 

PP       1 

N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8   9 

P.P. 

200 

30103 

125 

146 

168 

190 

211 

233 

255 

276  298 

20I 

320 

341 

363 

384 

406 

428 

471 

492  514 

22   21 

202 

53S 

557 

578 

600 

621 

643 

664 

\899 

707  728 

I 
2 
3 
4 

2/2    2,1 
4,4    4,2 

6,6   6,3 

8,8   8,4 

ii^o   10,5 

13,2   12,6 

15,4   14,7 
17,6   16,8 
19,8   18,9 

203 

750 

771 

792 

814 

835 

856 

878 

920  942 

204 

963 

984 

*oo6 

*027 

*048 

*o69 

*09i 

*II2 

*i33  *i54 

20s 

31  175 

197 

218 

239 

260 

281 

302 

323 

345  366 

206 

387 

408 

429 

450 

471 

492 

513 

534 

555  576 

207 

597 

618 

639 

660 

681 

702 

723 

744 

765  785 

7 
8 

208 

806 

827 

848 

869 

890 

911 

931 

952 

973  994 

209 

32015 

035 

056 

077 

098 

118 

139 

160 

181  201 

9 

210 

222 

243 

263 

284 

305 

32S 

346 

366 

387  408 

1 

211 

428 

449 

469 

490 

510 

531 

552 

572 

593  613 

20      1 

212 

634 

654 

675 

695 

715 

736 

756 

m 

797  818 

213 

838 

858 

879 

899 

919 

940 

960 

980 

^ifOoi  ^021 

I 

2,0 

214 
215 

33041 

062 

082 

102 

122 

143 

163 

183 

203  224 

2 

3 
4 

4,0 

6,0 

8,0. 

10  0 

244 

264 

284 

304 

325 

345 

365 

385 

405  425 

216 

445 

465 

486 

506 

526 

546 

566 

586 

606  626 

12  0 

217 

646 

666 

686 

706 

726 

746 

766 

786 

806  826 

7 
8 

14^0 
16  0 

218 

846 

866 

885 

905 

925 

945 

965 

985  »oo5  #025 

219 

34044 

064 

084 

IO.t__I24 

143 

163 

183 

203  223 

9 

18^0 

220 

242 

262 

282 

301 

321 

341 

361 

380 

400  420 

1 

221 

439 

459 

479 

498 

518 

537 

557 

577 

596  616 

in    1 

222 

635 

655 

674 

694 

713 

733 

753 

772 

792  811 

223 

830 

850 

869 

889 

908 

928 

947 

967 

986  *oo5 

I 

1,9 
3,8 

1% 

9,5 
11,4 
13,3 
15,2 
17,1 

224 

35025 

044 

064 

083 

102 

122 

141 

160 

180  199 

2 

3 
4 

7 
8 

9 

225 

218 

238 

257 

276 

295 

315 

334 

353 

372  392 

226 

411 

430 

449 

468 

488 

526 

545 

564  583 

227 

603 

622 

641 

660 

679 

698 

717 

736 

755  774 

228 

793 

813 

832 

851 

870 

889 

908 

927 

946  965 

229 

984  *oo3  ^021  ^040  *o59 

*078 

*097 

*ii6 

*i35  *i54 

230 

36173 

192 

211 

229 

248 

267 

286 

305 

324  342 

231 

361 

380 

399 

418 

436 

455 

474 

493 

511  530 

18 

232 

549 

568 

586 

605 

624 

642 

661 

680 

698  717 

I 
2 
3 

1,8 
3,6 

5,4 

233 
234 

736 
922 

754 
940 

773 
959 

791 

977 

810 
996 

829 
*oi4 

847 
*033 

866 

884  903 

*o5i  ^070  *o88 

235 

37  107 

125; 

144 

162 

181 

199 

218 

236 

254  273 

4 

7,2 

10,8 

236 

291 

310 

328 

346 

365 

383 

401 

420 

438  457 

237 

475 

493 

511 

530 

548 

566 

585 

603 

621  639 

I 

12  6 

238 

658 

676 

694 

712 

731 

749 

767 

785 

803  822 

\ii 

239 

840 

858 

876 

894 

912 

931 

949 

967 

985  *oo3 

9 

240 

38  021 

039 

057 

07S 

093 

112 

130 

148 

166  184 

1 

241 

202 

220 

238 

256 

274 

292 

310 

328 

346  364 

17      1 

242 

382 

399 

417 

435 

453 

471 

489 

507 

525  543 

243 

561 

578 

596 

614 

632 

650 

668 

686 

703  721 

I 

1,7 

244 

739 

757 

m 

792 

810 

828 

846 

863 

881  899 

2 

3 
4 
5 
6 

3,4 
5,1 

6,8 

8,5 
10  2 

245 

917 

934 

952 

970 

987 

*005  *023 

*04i 

#058  #076 

246 

39094 

III 

129 

146 

164 

182 

199 

217 

235  252 

247 

270 

287 

305 

322 

340 

358 

375 

393 

410  428 

I 

ii!9 
13,6 
15,3 

248 

44S 

463 

498 

51S 

533 

550 

568 

585  602 

249 

620 

637 

655 

672 

690 

707 

724 

742 

759  777 

9 

250 

794 

811 

829 

846 

863 

881 

898 

915 

933  950 

N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8   9 

P.P. 

N. 

L.  0 

1 

2 

3 

4 

5   6   7 

8 

9 

P.P. 

250 

39  794 

811 

829 

846 

863 

881  898  915 

933 

930 

251 

967 

985 

#002  ^^019  #037 

#054  #071  *o88 

*io6 

*I23 

18 

252 

40  140 

157 

175 

192 

209 

226  243  261 

278 

295 

1,8 
3,6 
5,4 

7,2 

253 

312 

329 

346 

364 

381 

398  415  432 

449 

466 

1 

254 

483 

500 

518 

535 

552 

569  586  603 

620 

637 

2 
3 

4 

I 

255 

654 

671 

688 

705 

722 

739  756  773 

790 

807 

256 

824 

841 

858 

875 

892 

909.  926  943 

960 

976 

9/0 
10  8 

^=>l 

993  *oio  *027 

#044^*061 

#078  »C395  *III 

*I28 

*i45 

7 
8 

I2'6 

14,4 
16,2 

258 

41  162 

179* 

196 

212 

229 

246  263  280 

296 

313 

259 
260 

330 

34r  363 

380 

397 

414  430  447 

464 

481 

9 

497 

514 

531 

547 

564 

581  597  614 

631 

647 

1 

261 

664 

681 

697 

714 

731 

747  764  780 

797 

814 

17     1 

262 

830 

847 

863 

880 

896 

913  929  946 

963 

979 

I 
2 
3 
4 

1,7 
3,4 

5,1 
6,8 

8,5 
10  2 

263 

996 

*OI2  #029  ^045 

*o62 

*078  *095  »iii 

*I27  *I44 

264 

42  160 

177 

193 

210 

226 

243  259  275 

292 

308 

26s 

325 

'341 

357 

37r 

390 

406  423  439 

45S 

472 

266 

488 

504 

521 

537 

553 

570  586  602 

619 

635 

267 

651 

684 

700 

716 

732  749  765 

781 

797 

7 
8 

11,9 
13,6 
15,3 

268 

813 

830 

846 

862 

878 

894  911  927 

943 

959 

269 

975 

991 

*oo8 

*024  »040 

#056  ,072  ,088 

#104 

*I20 

9 

270 

43  136 

152 

169 

185 

201 

217  233  249 

265 

281 

1 

271 

297 

313 

329 

345 

361 

377  393  409 

425 

441 

le    1 

272 

457 

473 

489 

505 

521 

537  553  569 

584 

600 

1,6 

273 

616 

632 

648 

664 

680 

696  712  727 

743 

759 

I 

274 

77^ 

791 

807 

823 

838 

854  870  886 

902 

917 

2 

3 
4 

8  0 

275 

933 

949 

965 

981 

996 

4>oi2  »028  #044  *o59 

*o7S 

276 

44091 

107 

122 

138 

154 

170  185  201 

217 

232 

9,6 
II  2 

277 

248 

264 

279 

295 

3" 

326  342  358 

373 

389 

I 

278 

404 

420 

436 

451 

467 

483  498  514 

529 

545 

12,8 

279 

560 

576 

592 

607 

623 

638  654  669 

685 

700 

9 

14^4 

280 

716 

731 

747 

762 

778 

793  809  824 

840 

855 

1 

281 

871 

886 

902 

917 

932 

948  963  979 

994  *oio 

15      1 

282 

45025 

040 

056 

071 

086 

102  117  133 

148 

163 

1,5 

283 

179 

194 

209 

225 

240 

255  271  286 

301 

317 

I 

284 

332 

347 

362 

378 

393 

408  423  439 

454 

469 

2 

3 
4 
5 
6 

3,0 
4,5 
6,0 
7,5 
9,0 
10,5 
12  0 

285 

484 

500 

515 

530 

545 

561  576  591 

606 

621 

286 

637 

652 

667 

682 

697 

712  728  743 

758 

773 

287 

788 

803 

818 

834 

849 

864  879  894 

909 

924 

7 
8 

288 

939 

954 

969 

984 

*ooo 

*oi5  *o3o  »o45 

*o6o 

*075 

289 
290 

46  090 

105 

120 

135 

150 

165  180  195 

210 

225 

9 

^3,S 

240 

255 

270 

285 

300 

315  330  345 

359 

374 

291 

389 

404 

419 

434 

449 

464  479  494 

509 

523 

14 

292 

538 

553 

568 

583 

598 

613  627  642 

657 

672 

293 

687 

716 

731 

746 

761  776  790 

805 

820 

I'  ■  ■ 

A, 4 
2,8 
4,2 

5,6 
7fi 
8,4 
9,8 
II  2 

294 

835 

850 

864 

879 

894 

909  923  938 

953 

967 

2 
3 
4 

5 
6 

295 

982 

997  *oi2 

»026 

»04i 

^^056  ifyjo  ip%t,  ^100  ^^114 

296 

47  129 

144 

159 

173 

188 

202  217  232 

246 

261 

297 

276 

290 

305 

319 

334 

349  363  378 

392 

407 

7 
3 

298 

422 

436 

451 

465 

480 

494  509  524 

538 

553 

299 

567 

582 

596 

611 

625 

640  654  669 

683 

698 

9 

12^6 

300 

712 

727 

741 

756 

770 

784  799  813 

828 

842 

N. 

L.  0 

1 

2 

3 

4 

5   6   7 

8 

9 

P.P. 

N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P.P. 

300 

47712 

727 

741 

756 

770 

784 

799 

813 

828 

842 

301 

857 

871 

885 

900 

914 

929 

943 

958 

972 

986 

302 

48  001 

oiS 

029 

044 

058 

073 

087 

lOI 

116 

130 

303 

144 

159 

173 

187 

202 

216 

230 

244 

259 

273 

15 

304 

287 

302 

316 

330 

344 

359 

373 

387 

401 

416 

305 

430 

444 

458 

473 

487 

501 

515 

530 

544 

558 

I  '^  '' 
2 

^/5 
3.0 

306 

572 

586 

601 

615 

629 

643 

657 

671 

686 

700 

3 

4,5 

307 

714 

728 

742 

756 

770 

785 

799 

813 

827 

841 

4 

6,0 

308 

855 

869 

883 

897 

911 

926 

940 

954 

968 

982 

5 

7,5 

309 

996 

^010  4|f024 

^038  ^052 

*o66 

*o8o 

*094 

*io8 

*I22 

7 

8 

9,0 
IO/5 
12,0 

310 

49136 

ISO 

164 

178 

192 

206 

220 

234 

248 

262 

311 

276 

290 

304 

318 

332 

346 

360 

374 

388 

402 

9 

13,5 

312 

415 

429 

443 

457 

471 

48s 

499 

513 

527 

541 

313 

554 

568 

582 

596 

610 

624 

638 

651 

679 

314 

693 

707 

721 

734 

748 

762 

776 

790 

803 

817 

14 

315 

831 

845 

859 

872 

886 

900 

914 

927 

941 

955 

316 

969 

982 

996 

*OIO  *024 

*o37  *o5i 

^065  ^079  ^092 

I 

1,4 
2  8 

317 

50  106 

120 

133 

147 

161 

174 

188 

202 

215 

229 

2 

318 

243 

256 

270 

284 

297 

311 

325 

338 

352 

365 

3 
4 
5 
6 

7,0 

8,4 

319 

379 

393 

406 

420 

433 

447 

461 

474 

488 

501 

320 

•.   515 

529 

542 

556 

569'. 

,  583 

596 

610 

623 

637 

321 

651 

664 

678 

691 

705 

718 

732 

745 

759 

772 

7 

9,8 

322 

786 

799 

813 

826 

840 

853 

866 

880 

893 

907 

8 

11,2 

323 

920 

934 

947 

961 

974 

987  ^001 

*oi4 

*028 

*04i 

9 

12.6 

324 

51055 

068 

081 

095 

108 

121 

135 

148 

162 

175 

325 

188 

202 

215 

228 

242 

255 

268 

282 

295 

308 

326 

322 

33S 

348 

362 

375 

388 

402 

415 

428 

441 

327 

45S 

468 

481 

495 

508 

521 

534 

548 

561 

574 

13 

328 

587 

601 

614 

627 

640 

654 

667 

680 

693 

706 

329 

720 

733 

746 

759 

772 

786 

799 

812 

825 

838 

I 
2 
3 

3,9 

330 

851 

865 

878 

891 

904 

917 

930 

943 

957 

970 

331 

983 

996 

*009  *022  *03^ 

*048 

*o6i 

*o75 

*o88 

*ioi 

4 

5,2 

332 

52114 

127 

140 

153 

166 

179 

192 

205 

218 

231 

6 

333 

244 

257 

270 

284 

297 

310 

323 

336 

349 

362 

334 

375 

388 

401 

414 

427 

440 

453 

466 

479 

492 

I 
9 

9,1 
10,4 

11,7 

335 

504 

517 

530 

543 

556 

569 

582 

595 

608 

621 

336 

634 

647 

660 

673 

686 

699 

711 

724 

737 

750 

337 

763 

776 

789 

802 

815 

827 

840 

853 

866 

879 

338 

892 

905 

917 

930 

943 

956 

969 

982 

994  *oo7 

339 

53020 

033 

046 

058 

071 

084 

097 

no 

122 

135 

10 

340 

148 

161 

173 

186 

199 

212 

224 

237 

250 

263 

I 

1,2 

341 

27S 

288 

301 

314 

326 

339 

352 

364 

377 

390 

2 

2,4 

342 

403 

415 

428 

441 

453 

466 

479 

491 

504 

517 

3 

36 

343 

529 

555 

567 

580 

593 

605 

618 

631 

643 

4 

4,8 

344 

656 

668 

681 

694 

706 

719 

732 

744 

757 

769 

6,0 

7,2 

8,4 

345 

782 

794 

807 

820 

832 

845 

857 

870 

882 

895 

346 

908 

920 

933 

945 

958 

970 

983 

995 

*oo8 

^020 

8 

9,6 

347 

54033 

04S 

058 

070 

083 

095 

108 

120 

133 

145 

9 

10,8 

348 

170 

183 

195 

208 

220 

233 

245 

258 

270 

349 

283 

295 

307 

320 

332 

345 

357 

370 

382 

394 

350 

407 

419 

432 

444 

456 

469 

481 

494 

506 

S18 

N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P.P. 

N. 

L.  0 

12   3   4 

5   6 

7 

8   9 

P.P. 

350 

54407 

419  432  444  456 

469  481 

494 

506  518 

351 

531 

543  555  568  580 

593  605 

617 

630  642 

352 

654 

667  679  691  704 

716  728 

741 

753  765 

353 

777 

790  802  814  827 

839  851 

864 

876  888 

354 

900 

913  925  937  949 

962  974 

986 

998  ^01 I 

I 
2 

13 

355 

55023 

035  047  060  072 

084  096 

108 

121  133 

356 

145 

157  169  182  194 

206  218 

230 

242  255 

3 

3.9 

357 

267 

279  291  303  315 

328  340 

352 

364  376 

4 

5,2 

358 

388 

400  413  425  437 

449  461 

473 

485  497 

5 

6,5 

359 

509 

522  534  546  558 

570  582 

594 

606  618 

7 
8 

7,8 
9,1 
10,4 

360 

630 

642  654  666  678 

691  703 

715 

727  739 

361 

751 

763  775  787  799 

811  .823 

835 

847  859 

9 

"/7 

362 

871 

883  895  907  919 

93/943 

955 

967  979 

3^3 

991 

^003  ^015  *o27  j^o38 

*oj^  *o62 

*074 

*o86  ^098 

364 

56  no 

122  134  146  158 

17b'"  182 

194 

205  217 

12 

365 

229 

241  253  265  277 

289  301 

312 

324  336 

366 

348 

360  372  384  396 

407  419 

431 

443  455 

I 

I  2 

367 

467 

478  490  502  514 

526  538 

561  573 

4,8 
6,0 

7,2 

368 

585 

597  608  620  632 

644  656 

667 

679  691 

3 
4 
5 
6 

369 

703 

714  726  738  750 

761  773 

785 

797  808 

370 

820 

832  844  855  867 

879  891 

902 

914  926 

371 

937 

949  961  972  984 

996  #008 

*oi9  ^031  *043 

7 

8,4 

372 

57054 

066  078  089  lOI 

113  124 

136 

148  159 

8 

9,6 

373 

^v- 

183  194  206  217 

229  241 

252 

264  276 

9 

10.8 

374 

287 

299  310  322  334 

34S  357 

368 

380  392 

375 

403 

415  426  438  449 

461  473 

484 

496  507 

376 

519 

530  542  553  565 

576  588 

600 

611  623 

377 

634 

646  657  669  680 

692  703 

715 

726  738 

11 

378 

749 

761  772  784  795 

807  818 

830 

841  852 

379 

864 

875  887  898  910 

921  933 

944 

95S  967 

I 
2 
3 

1,1 
2,2 

3,3 

380 

978 

990  *ooi  *oi3  *o^ 

*035  *047  *o58  *070  *o8i 

381 

58092 

104  115  127  138 

149  161 

172 

184  195 

4 

4,4 

382 

206 

218  229  240  252 

263  274 

286 

297  309 

5 

5,5 

383 

320 

331  343  354  36^ 

377  388 

399 

410  422 

6 

6/6 

384 

433 

444  456  467  478 

490  501 

512 

524  535 

I 
9 

7J 
8,8 
9,9 

385 

546 

557  569  580  591 

602  614 

625 

636  647 

386 

659 

670  681  692  704 

715  726 

737 

749  760 

387 

771 

782  794  805  816 

827  838 

850 

861  872 

388 

883 

894  906  917  928 

939  950 

961 

973  984 

389 

995 

^006  *oi7  *028  #040 

*o5i  *o62 

*073 

^084  #095 

in 

390 

59106 

118  129  140  151 

162  173 

184 

195  207 

I 

1,0 

391 

218 

229  240  251  262 

273  284 

295 

306  318 

2 

2,0 

392 

329 

340  351  362  373 

384  395 

406 

417  428 

3 

3,0 

393 

439 

450  461  472  483 

494  506 

517 

528  539 

4 

4,0 

394 

550 

561  572  583  594 

605  616 

627 

638  649 

7 

5,0 
6,0 
7,0 

395 

660 

671  682  693  704 

715  726 

737 

748  759 

396 

770 

780  791  802  813 

824  83S 

846 

857  868 

8 

8,0 

397 

879 

890  901  912  923 

934  945 

956 

966  977 

9 

9,0 

398 

988 

999  *OIO  *02I  ^032 

*043  *o54 

*o65 

^076  *o86 

399 

60097 

108   119   130   141 

152  T63 

173 

184  19S 

400 

206 

217   228   239   249 

260  271 

282 

293  304 

N. 

L.  0 

12   3   4 

5   6 

7 

8   9 

P.P. 

N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P.P. 

400 

60  206 

217 

228 

239 

249 

260 

271 

282 

293 

304 

401 

314 

325 

336 

347 

358 

369 

379 

390 

401 

412 

402 

423 

433 

444 

455 

466 

477 

487 

498 

509 

520 

403 

531 

541 

552 

563 

574 

584 

595 

606 

617 

627 

404 

638 

649 

660 

670 

681 

692 

703 

713 

724 

735 

405 

746 

756 

767 

778 

788 

799 

810 

821 

831 

842 

406 

853 

863 

874 

885 

895 

906 

917 

927 

938 

949 

11 

407 

959 

970 

981 

991 

*002 

*oi3  *o23  *o34  ^045  *o55 

408 

61066 

077 

087 

098 

109 

119 

130 

140 

151 

162 

I 

1,1 

409 

172 

183 

194 

204 

215 

225 

236 

247 

257 

268 

2 

3 
4 

2/2 

3.3 

4/4 

410 

278 

289 

300 

310 

321 

331 

342 

352 

363 

374 

411 

384 

395 

405 

416 

426 

437 

448 

458 

469 

479 

412 

490 

500 

511 

521 

532 

542 

553 

563 

574 

584 

I 

7,7 
8/8 

413 

595 

606 

616 

627 

637 

648 

658 

669 

679 

690 

414 

700 

711 

721 

73^ 

742 

752 

763 

773 

784 

794 

9 

9^9 

415 

805 

815 

826 

836 

847 

857 

868 

878 

888 

899 

41b 

909 

920 

930 

941 

951 

962 

972 

982 

993 

*oo3 

417 

62  014 

024 

034 

045 

055 

066 

C76 

086 

097 

107 

418 

118 

128 

138 

149 

159 

170 

180 

190 

20  r 

211 

419 

221 

232 

242 

252 

263 

273 

284 

294 

304 

315 

420 

325 

335 

346 

356 

366 

377 

387 

397 

408 

418 

421 

428 

439 

449 

459 

469 

480 

490 

500 

511 

521 

10 

422 

531 

542 

552 

562 

572 

583 

593 

603 

613 

624 

423 

634 

644 

655 

665 

675 

685 

696 

706 

716 

726 

I ' "  - 

1,0 

424 

737 

747 

757 

767 

778 

788 

798 

808 

818 

829 

2 
3 
4 

2/0 

3/0 
4/0 
5/0 
6  0 

425 

839 

849 

859 

870 

880 

890 

900 

910 

921 

931 

426 

941 

951 

961 

972 

982 

992  5,f002  ^012  *022  j,j033 

427 

63043 

053 

063 

073 

083 

094 

104 

114 

124 

134 

7 
8 

9 

7^0 
8/O 
9/0 

428 

144 

155 

165 

175 

185 

195 

205 

215 

225 

236 

429 

246 

256 

266 

276 

286 

296 

306 

317 

327 

337 

430 

347 

357 

367 

377 

387 

397 

407 

417 

428 

438 

431 

448 

458 

468 

478 

488 

498 

508 

518 

528 

538 

432 

548 

558 

568 

579 

589 

599 

609 

619 

629 

639 

433 

649 

659 

669 

679 

689 

699 

709 

719 

729 

739 

434 

749 

759 

769 

779 

789 

799 

809 

819 

829 

839 

435' 

849 

859 

869 

879 

889 

899 

909 

919 

929 

939 

436 

949 

959 

969 

979 

988 

998 

*oo8 

*oi8 

*028 

*o38 

9 

437 

64  048 

058 

068 

078 

088 

098 

108 

118 

128 

137 

438 

147 

157 

167 

177 

187 

197 

207 

217 

227 

237 

I 

0/9 

439 

246 

256 

266 

276 

286 

296 

306 

316 

326 

335 

2 
3 
4 

1/8 

2/7 

3/6 

4/5 

440 

34S 

355 

365 

375 

385 

395 

404 

414 

424 

434 

441 

444 

454 

464 

473 

483 

493 

503 

513 

523 

532 

442 

542 

552 

572 

582 

591 

601 

611 

621 

631 

5/4 
6/3 

443 

640 

650 

660 

670 

680 

689 

699 

709 

719 

729 

7 
8 

9 

444 

738 

748 

758 

768 

777 

787 

797 

807 

816 

826 

7/2 

8/1 

445 

836 

846 

856 

865 

875 

885 

895 

904 

914 

924 

446 

933 

943 

953 

963 

972 

982 

992  *002  *OII 

*02I 

447 

65031 

040 

050 

060 

070 

079 

089 

099 

108 

118 

448 

128 

137 

147 

157 

167 

176 

186 

196 

205 

215 

449 

225 

234 

244 

254 

263 

273 

283 

292 

302 

312 

450 

321 

331 

341 

350 

360 

369 

379 

389 

398 

408 

N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P.P. 

N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P.P. 

450 

65321 

331 

341 

350 

360 

369 

379 

389 

398 

408 

451 

418 

427 

437 

447 

456 

466 

475 

485 

495 

504 

452 

514 

523 

533 

543 

552 

562 

571 

581 

591 

600 

453 

610 

619 

629 

^39 

648 

658 

667 

677 

696 

454 

706 

71S 

725 

734 

744 

753 

763 

772 

782 

792 

455 

801 

811 

820 

830 

839 

849 

858 

868 

877 

887 

456 

896 

906 

916 

925 

935 

944 

954 

963 

973 

982 

10 

457 

992  ^^OOI  *OII 

*020 

*o3o 

5^039  ^049  ^058 

*o68 

*077 

458 

66087 

096 

106 

115 

124 

134 

143 

153 

162 

172 

I 

^P 

459 

181 

191 

200 

210 

219 

229 

238 

247 

257 

266 

2 

3 

4 

5 
6 

2,0 

3,0 
4.0 

5,0 
6  0 

460 

276 

28S 

295 

304 

314 

323 

332 

342 

351 

361 

461 

370 

380 

389 

398 

408 

417 

427 

436 

445 

455 

462 

464 

474 

483 

492 

502 

511 

521 

530 

539 

549 

7 
8 

7!o 
8  0 

463 

558 

567 

577 

596 

605 

614 

624 

633 

642 

464 

652 

661 

671 

680 

689 

699 

708 

717 

727 

736 

9 

9!° 

465 

745 

755 

764 

773 

783 

792 

801 

811 

820 

829 

466 

839 

848 

857 

867 

876 

88^ 

894 

904 

913 

922 

467 

932 

941 

950 

960 

969 

978 

987 

997 

*oo6 

*oi5 

468 

67025 

034 

043 

052 

062 

071 

080 

089 

099 

108 

469 

117 

127 

136 

145 

154 

164 

173 

182 

191 

201 

470 

210 

219 

228 

237 

247 

256 

265 

274 

284 

293 

471 

302 

311 

321 

330 

339 

348 

357 

367 

376 

385 

9 

472 

394 

403 

413 

422 

431 

440 

449 

459 

.468 

477 

473 

486 

495 

504 

514 

523 

532 

541 

550 

560 

569 

I 

",9 
1.8 

4:5 

'5.4 
8.1 

474 

578 

587 

596 

605 

614 

624 

633 

642 

651 

660 

2 
3 

4 
5 
6 

7 
8 

9 

475 

669 

679 

688 

697 

706 

715 

724 

733 

742 

752 

476 

761 

770 

779 

788 

797 

806 

815 

825 

834 

843 

477 

852 

861 

870 

879 

888 

897 

906 

916 

925 

934 

478 

943 

952 

961 

970 

979 

988 

997 

*oo6 

*oi5  *024 

479 

68034 

043 

052 

061 

070 

079 

088 

097 

106 

"5 

480 

124 

133 

142 

151 

160 

169 

178 

187 

196 

205 

481 

215 

224 

233 

242 

251 

260 

269 

278 

287 

296 

482 

305 

314 

323 

332 

341 

350 

359 

368 

377 

386 

483 

395 

404 

413 

422 

431 

440 

449 

458 

467 

476 

484 

485 

494 

502 

511 

520 

529 

538 

547 

556 

565 

485 

574 

583 

592 

601 

610 

619 

628 

637 

646 

655 

486 

^4 
-7^ 

673 

681 

690 

699 

708 

717 

726 

735 

744 

R 

487 

762 

771 

780 

789 

797 

806 

815 

824 

833 

0.8 

2.4 
3,2 

488 

842 

851 

860 

869 

878 

886 

895. 

904 

913 

922 

I 

489 

931 

940 

949 

958 

966 

975 

984 

993  *oo2 

*oii 

2 

3 
4 

490 

69  020 

028 

037 

046 

05s 

064 

073 

082 

090 

099 

491 

108 

117 

126 

135 

144 

152 

161 

170 

179 

188 

5 
6 

7 
8 

4/0 
4.8 
5,6 
6,4 
7,2 

492 

197 

205 

214 

223 

232 

241 

249 

258 

267 

276 

493 

285 

294 

302 

311 

320 

329 

338 

346 

355 

364 

494 

373 

381 

390 

399 

408 

417 

42S 

434 

443 

452 

9 

495 

461 

469 

478 

487 

496 

504 

513 

522 

531 

539 

496 

548 

557 

566 

574 

583 

592 

601 

609 

618 

627 

497 

636 

644 

653 

662 

671 

679 

688 

697 

705 

714 

498 

723 

732 

740 

749 

758 

767 

ITS 

784 

793 

801 

499 

810 

819 

827 

836 

845 

854 

862 

871 

880 

888 

500 

897 

906 

914 

923 

932 

940 

949 

958 

966 

975 

N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P.P. 

N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8   9 

P.P. 

500 

69897 

906 

914 

923 

932 

940 

949 

958 

966  975 

501 

992  j^OOI  *OIO 

*oi8 

*027 

*036 

*o44  *o53  *o62 

502 

70  070 

079 

088 

% 

105 

114 

122 

131 

140  148 

503 

157 

165 

174 

191 

200 

209 

217 

226  234 

504 

243 

252 

260 

269 

278 

286 

295 

303 

312  3a.; 

505 

329 

338 

346 

355 

364 

372 

381 

389 

398  406 

506 

415 

424 

432 

441 

449 

458 

467 

475 

484  492 

Q 

507 

501 

509 

518 

526 

535 

544 

552 

561 

569  578 

S08 

586 

595 

603 

612 

621 

629 

638 

646 

655  663 

I 

0/9 
1/8 
2.7 
3,6 
4/5 
5/4 
6/3 

7/2 

8/1 

509 

672 

680 

689 

697 

706 

714 

723 

731 

740  749 

2 
3 

4 
5 
6 

I 

510 

757 

766 

774 

783 

791 

800 

808 

817 

825  834 

511 

842 

851 

859 

868 

876 

885 

893 

902 

910  919 

512 

927 

935 

944 

952 

961 

969 

978 

986 

995  *oo3 

513 

71  012 

020 

029 

037 

046 

054 

063 

071 

079  088 

514 

096 

105 

"3 

122 

130 

139 

147 

15s 

164  172 

9 

515 

181 

189 

198 

206 

214 

223 

231 

240 

248  257 

516 

265 

273 

282 

290 

299 

307 

31S 

324 

332  341 

517 

349 

357 

366 

374 

383 

391 

399 

408 

416  425 

518 

433 

441 

450 

458 

466 

475 

483 

492 

500  508 

519 

517 

525 

533 

542 

550 

559 

567 

575 

584  592 

520 

600 

609 

617 

625 

634 

642 

650 

659 

667  675 

521 

684 

692 

700 

709 

717 

725 

734 

742 

750  759 

8 

522 

767 

11^ 

784 

792 

800 

809 

817 

825 

834  842 

523 

850 

858 

867 

875 

883 

892 

900 

908 

917  925 

If '  " 

0,0 
1/6 

2/4 
3/2 

524 

933 

941 

950 

958 

966 

975 

983 

991 

999  *oo8 

2 
3 
4 

525 

72016 

024 

032 

041 

049 

057 

066 

074 

082  090 

526 

099 

107 

115 

123 

132 

140 

148 

156 

165  173 

4'° 
4/8 
5/6 
6/4 

7/2 

527 

181 

189 

198 

206 

214 

222 

230 

239 

247  25s 

528 

263 

272 

280 

288 

296 

304 

313 

321 

329  337 

9 

529 

346 

354 

362 

370 

378 

387 

395 

403 

411  419 

530 

428 

436 

444 

452 

460 

469 

477 

485 

493  501 

531 

509 

518 

526 

534 

542 

550 

558 

567 

575  583 

532 

591 

599 

607 

616 

624 

632 

640 

648 

656  66^ 

533 

673 

681 

689 

697 

705 

713 

722 

730 

738  746 

534 

754 

762 

770 

779 

787 

795 

803 

811 

819  827 

535 

835 

843 

852 

860 

868 

876 

884 

892 

900  908 

536 

916 

925 

933 

941 

949 

957 

965 

973 

981  989 

7 

537 

997 

*oo6 

*oi4  *022  ^030 

*038  *046 

*o54 

Jif062  ^070 

538 

73078 

086 

094 

102 

III 

119 

127 

135 

143   151 

I 

0/7 

539 

159 

167 

175 

183 

191 

199 

207 

215 

223  231 

2 
3 

4 

1/4 

2/1 
2/8 

540 

239 

247 

25s 

263 

272 

280 

288 

296 

304  312 

541 

320 

328 

344 

352 

360 

368 

376 

384  392 

5 
6 

3/5 

542 

400 

408 

416 

i^ 

432 

440 

448 

456 

464  472 

4/2 

543 

480 

488 

496 

504 

512 

520 

528 

536 

544  552 

7 
8 

9 

5/6 
6/3 

544 

560 

568 

576 

584 

592 

600 

608 

616 

624  632 

545 

640 

648 

656 

664 

672 

679 

687 

695 

703  7" 

546 

719 

727 

735 

743 

751 

759 

767 

77.5 

783  791 

547 

799 

807 

815 

823 

830 

838 

846 

854 

862  870 

548 

878 

886 

894 

902 

910 

918 

926 

933 

941  949 

549 

957 

965 

973 

981 

989 

997  *ooS  *oi3  *020  #028 

550 

74036 

044 

052 

060 

068 

076 

084 

092 

099  107 

^ 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8   9 

P.P. 

N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8   9 

P.P. 

550 

74036 

044 

052 

060 

068 

076 

084 

092 

099  107 

551 

115 

123 

131 

139 

147 

155 

162 

170 

178  186 

552 

194 

202 

210 

218 

225 

233 

241 

249 

257  265 

553 

273 

280 

288 

296 

304 

312 

320 

327 

335  343 

554 

351 

359 

367 

374 

382 

390 

398 

406 

414  421 

555 

429 

437 

445 

453 

461 

468 

476 

484 

492  500 

556 

507 

515 

523 

531 

539 

547 

554 

562 

570  578 

557 

586 

593 

601 

609 

617 

624 

632 

640 

648  656 

558 

663 

671 

679 

687 

695 

702 

710 

718 

726  733 

559 

741 

749 

757 

764 

772 

780 

788 

796 

803  811 

560 

819 

827 

834 

842 

850 

858 

865 

873 

881  889 

561 

896 

904 

912 

920. 

927 

935 

943 

950 

958  966 

8 

562 

974 

981 

989 

997  *oo5  1 

^012  ^020 

*028 

*03S  *043 

I 

Iy6 

2,4 
3.2 

4,0 

4.8 
SA 
6,4 

7/2. 

563 

75051 

059 

066 

074 

082 

089 

097 

105 

113  120 

564 

128 

136 

143 

151 

159 

166 

174 

182 

189  197 

2 

3 
4 
5 
6 

565 

205 

213 

220 

228 

236 

243 

251 

259 

266  274 

566 

282 

289 

297 

305 

312 

320 

328 

335 

343  351 

567 

358 

366 

374 

381 

389 

397 

404 

412 

420  427 

I 

568 

435 

442 

450 

458 

465 

473 

481 

488 

496  504 

569 

5" 

519 

526 

534 

542 

549 

557 

565 

572  580 

9 

570 

587 

595 

603 

610 

618 

626 

633 

641 

648  656 

571 

664 

671 

679 

686 

694 

702 

709 

■  717 

724  732 

572 

740 

747 

755 

762 

770 

778 

785 

793 

800  808 

573 

815 

823 

831 

838 

846 

853 

861 

868 

876  884 

574 

891 

899 

906 

914 

921 

929 

937 

944 

952  959 

575 

967 

974 

982 

989 

997 

*005  *OI2  *020  »027  *035 

576 

76042 

050 

057 

065 

072 

080 

087 

095 

103  no 

577 

118 

125 

140 

148 

155 

163 

170 

178  185 

578 

193 

200 

208 

215 

223 

230 

238 

24S 

253  260 

579 

268 

275 

283 

290 

298 

305 

313 

320 

328  33S 

580 

343 

350 

358 

365 

373 

380 

388 

395 

403  410 

581 

418 

425 

433 

440 

448 

455 

462 

470 

477  485 

7 

582 

492 

500 

507 

515 

522 

530 

537 

545 

552  559 

583 

567 

574 

582 

589 

597 

604 

612 

619 

626  634 

I 

0/7 

584 

641 

649 

656 

664 

671 

678 

686 

693 

701  708 

2 

3 
4 

1/4 
2,1 

2/8 

3/5 

4/2 

P 

6/3 

585 

716 

723 

730 

738 

745 

753 

760 

768 

775  782 

586 

790 

797 

805 

812 

819 

827 

834 

842 

849  856 

587 

864 

871 

879 

886 

893 

901 

908 

916 

923  930 

I 

588 

938 

945 

953 

960 

967 

975 

982 

989 

997  *oo4 

589 

77012 

019 

026 

034 

041 

048 

056 

063 

070  078 

9 

590 

085 

093 

100 

107 

115 

122 

129 

137 

144  151 

591 

159 

166 

173 

181 

188 

195 

203 

210 

217  225 

592 

232 

240 

247 

254 

262 

269 

276 

283 

291  298 

593 

305 

313 

320 

327 

335 

342 

349 

357 

364  371 

594 

595 

379 

386 

393 

401 

408 

415 

422 

430 

437  444 

452 

459 

466 

474 

481 

488 

495 

503 

510  517 

596 

525 

532 

539 

546 

554 

561 

568 

576 

583  590 

597 

597 

605 

612 

619 

627 

634 

641 

648 

656  663 

598 

670 

677 

685 

692 

699 

706 

714 

721 

728  735 

599 

743 

750 

757 

764 

772 

779 

786 

793 

801  808 

600 

815 

822 

830 

837 

844 

851 

859 

866 

873  880 

N. 

L.  0 

1 

2 

3 

4 

6 

6 

7 

8   9 

P.P. 

N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P.P. 

600 

77  8iS 

822 

830 

837 

844 

851 

859 

866 

873 

880 

6oi 

887 

895 

902 

909 

916 

924 

931 

938 

945 

952 

602 

960 

967 

974 

981 

988 

996  ^003  *oio  ^017  ^025 

603 

78  032 

039 

046 

053 

061 

068 

075 

082 

089 

097 

004 

104 

III 

118 

125 

132 

140 

147 

154 

161 

168 

60s 

176 

183 

190 

197 

204 

211 

219 

226 

233 

240 

606 

247 

254 

262 

269 

276 

283 

290 

297 

305 

312 

8 

607 

319 

326 

333 

340 

347 

355 

362 

369 

376 

383 

608 

390 

398 

405 

412 

419 

426 

433 

440 

447 

455 

I 

0^0 
2.4 

3,2 

609 

462 

469 

476 

483 

490 

497 

504 

512 

519 

526 

2 

3 
4 

610 

533 

540 

547 

554 

561 

569 

576 

583 

590 

597 

611 

604 

611 

618 

625 

633 

640 

647 

654 

661 

668 

6 

4/0 

5/6 
6/4 

7/2 

612 

675 

682 

689 

696 

704 

711 

718 

725 

732 

739 

613 

753 

760 

767 

774 

781 

789 

796 

803 

810 

I 
9 

614 

817 

824 

831 

838 

84s 

852 

859 

866 

873 

880 

615 

888 

895 

902 

909 

916 

923 

930 

937 

944 

951 

616 

958 

96| 

972 

979 

986 

993  *ooo  *oo7  *oi4  *02i 

617 

79029 

036 

043 

050 

057 

064 

071 

078 

085 

092 

618 

099 

106 

113 

120 

127 

134 

141 

148 

155 

162 

619 

169 

176 

183 

190 

197 

204 

211 

218 

225 

232 

620 

239 

246 

253 

260 

267 

274 

281 

288 

295 

302 

621 

309 

316 

323 

330 

337 

344 

351 

358 

365 

372 

7 

622 

379 

386 

393 

400 

407 

414 

421 

428 

435 

442 

623 

449 

456 

463 

470 

477 

484 

491 

498 

505 

511 

I 

0/7 

624 

518 

52s 

532 

539 

546 

553 

560 

567 

574 

581 

2 

3 
.4 

1/4 

2/1 
2/8 

625 

588 

595 

602 

609 

616 

623 

630 

637 

644 

650 

626 

657 

664 

671 

678 

68S 

692 

699 

706 

713 

720 

6 

3/5 

627 

727 

734 

741 

748 

754 

761 

768 

71^ 

782 

789 

4/2 

628 

796 

803 

810 

817 

824 

831 

837 

844 

851 

858 

7 
8 

9 

4/9. 

^/ 
6.3 

629 

865 

872 

879 

886 

893 

900 

906 

913 

920 

927 

630 

934 

941 

948 

955 

962 

969 

975 

982 

989 

996 

631 

80003 

010 

017 

024 

030 

037 

044 

051 

058 

065 

632 

072 

079 

085 

092 

099 

106 

113 

120 

127 

134 

633 

140 

147 

154 

161 

168 

175 

182 

188 

195 

202 

634 

209 

216 

223 

229 

236 

243 

250 

257 

264 

271 

63s 

277 

284 

291 

298 

305 

312 

318 

325 

332 

339 

636 

346 

353 

359 

366 

373 

380 

387 

393 

400 

407 

R. 

637 

414 

421 

428 

434 

441 

448 

455 

462 

468 

475 

v» 

638 

482 

489 

496 

502 

509 

516 

523 

530 

536 

543 

I 

0,6 

639 

5So 

557 

564 

570 

577 

584 

591 

598 

604 

611 

2 

3 
4 

1/2 
1/8 

2/4 

640 

618 

625 

632 

638 

645 

652 

659 

665 

672 

679 

641 

686 

693 

699 

706 

713 

720 

726 

733 

740 

747 

6 

3'? 

3/6 

642 

754 

760 

767 

774 

781 

787 

794 

801 

808 

814 

643 

821 

828 

835 

841 

848 

855 

862 

868 

875 

882 

7 
8 

9 

4/2 

4/8 

5/4 

644 

889 

895 

902 

909 

916 

922 

929 

936 

943 

949 

645 

956 

963 

969 

976 

983 

990 

996  ^003  *OIO  415OI7 

646 

81023 

030 

037 

043 

0^0 

057 

064 

070 

077 

084 

647 

090 

097 

104 

III 

117 

124 

131 

137 

144 

151 

648 

158 

164 

171 

178 

184 

191 

198 

204 

211 

218 

649 

224 

231 

238 

245 

251 

258 

265 

271 

278 

285 

650 

291 

298 

305 

3" 

318 

325 

331 

338 

345 

351 

N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P.P. 

N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P.P. 

650 

81  291 

298 

305 

311 

318 

325 

331 

338 

345 

351 

651 

358 

365 

371 

378 

385 

391 

398 

405 

411 

418 

652 

425 

43^ 

438 

445 

45^ 

458 

465 

471 

478 

485 

653 

491 

498 

505 

511 

518 

525 

531 

538 

544 

551 

654 

558 

564 

571 

578 

584 

591 

598 

604 

611 

617 

655 

624 

631 

637 

644 

651 

657 

664 

671 

677 

684 

656 

690 

697 

704 

710 

717 

723 

730 

737 

743 

750 

657 

757 

763 

770 

776 

783 

790 

796 

803 

809 

816 

658 

823 

829 

836 

842 

849 

856 

862 

869 

875 

882 

659 

889 

895 

902 

908 

915 

921 

928 

935 

941 

948 

660 

0  954 

961 

968 

974 

981 

987 

994  *ooo  #007  *oi4 

661 

82020 

027 

033 

040 

046 

053 

060 

066 

073 

079 

7 

662 

086 

092 

099 

105 

112 

119 

125 

132 

138 

145 

663 

151 

158 

164 

171 

178 

184 

191 

197 

204 

210 

I 

0,7 

664 

217 

223 

230 

236 

243 

,249 

256 

263 

269 

276 

2 

3 
4 

1/4 

2/1 

2,8 
3/5 

4/2 

4/9 

^'^ 
6/3 

665 

282 

289 

295 

302 

308 

315 

321 

328 

334 

341 

666 

347 

354 

360 

367 

373 

380 

387 

393 

400 

406 

667 

413 

419 

426 

432 

439 

445 

452 

458 

465 

471 

7 
8 

668 

478 

484 

491 

497 

504 

510 

517 

523 

530 

536 

669 

543 

549 

556 

562 

569 

575 

582 

588 

595 

601 

9 

670 

607 

614 

620 

627 

633 

640 

646 

653 

659 

666 

671 

672 

679 

685 

692 

698 

705 

711 

718 

724 

730 

672 

737 

743 

750 

756 

763 

769 

776 

782 

789 

795 

673 

802 

808 

814 

821 

827 

834 

840 

847 

853 

860 

674 

866 

872 

879 

885 

892 

898 

905 

911 

918 

924 

675 

930 

937 

943 

950 

956 

963 

969 

975 

982 

988 

676 

995 

*ooi 

*oo8 

^014  j|t020 

*027  *033  ^040 

#046 

*052 

677 

83059 

065 

072 

078 

085 

091 

097 

104 

no 

117 

678 

123 

129 

136 

142 

149 

155 

161 

168 

174 

181 

679 

187 

193 

200 

206 

213 

219 

225 

232 

238 

245 

680 

251 

257 

264 

270 

276 

283 

289 

296 

302 

308 

681 

315 

321 

327 

334 

340 

347 

353 

359 

366 

372 

R 

682 

378 

385 

391 

398 

404 

410 

417 

423 

429 

436 

O/6 

683 

442 

448 

455 

461 

467 

474 

480 

487 

493 

499 

I 

684 

506 

512 

518 

525 

531 

537 

544 

550 

563 

2 

3 
4 

1/2 

1/8 

2/4 

685 

569 

575 

582 

588 

594 

601 

607 

613 

620 

626 

686 

632 

639 

645 

651 

658 

664 

670 

677 

683 

689 

5 
6 

3/0 
3/6 

687 

696 

702 

708 

715 

721 

727 

734 

740 

746 

753 

688 

759 

765 

771 

778 

784 

790 

797 

803 

809 

816 

7 
8 

9 

4/2 

4/8 
5/4 

689 

822 

828 

835 

841 

847 

853 

860 

866 

872 

879 

690 

885 

891 

897 

904 

910 

916 

923 

929 

935 

942 

691 

948 

954 

960 

967 

973 

979 

985 

992 

998  ^004 

6q2 

84  on 

017 

023 

029 

036 

042 

048 

055 

061 

067 

693 

073 

080 

086 

4)2 

098 

105 

III 

117 

123 

130 

694 
695 

136 

142 

148 

155 

161 

167 

173 

180 

186 

192 

198 

205 

211 

217 

223 

230 

236 

242 

248 

255 

696 

261 

267 

273 

280 

286 

292 

298 

305 

311 

317 

697 

323 

330 

336 

342 

348 

354 

361 

367 

373 

379 

698 

386 

392 

398 

404 

410 

417 

423 

429 

435 

442 

699 

448 

454 

460 

466 

473 

479 

485 

491 

497 

504 

700 

510 

516 

522 

528 

535 

541 

547 

553 

559 

566 

N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P.P. 

N. 

L.  0 

1 

2 

3 

4 

6 

6 

7 

8 

9 

P.P. 

700 

84510 

516 

522 

528 

535 

541 

547 

553 

559 

566 

701 

572 

578 

584 

590 

597 

603 

609 

615 

621 

628 

702 

634 

640 

646 

652 

658 

665 

671 

677 

683 

689 

703 

696 

702 

708 

714 

720 

726 

733 

739 

745 

751 

704 

757 

763 

770 

776 

782 

788 

794 

800 

807 

813 

705 

819 

825 

831 

837 

844 

850 

856 

862 

868 

874 

706 

880 

887 

893 

899 

90s 

911 

917 

924 

930 

936 

7 

707 

942 

948 

954 

960 

973 

979 

985 

991 

997 

T 

0,7 
1,4 

2/1 

2.8 

3/S 

708 

85003 

009 

016 

022 

028 

034 

040 

046 

052 

058 

2 

709 

065 

071 

077 

083 

089 

095 

lOI 

107 

114 

120 

3 
4 

5 

710 

126 

132 

138 

144 

150 

156 

163 

169 

'71 

181 

711 

187 

193 

199 

20S 

211 

217 

224 

230 

236 

242 

6 

4/2 

712 

248 

254 

260 

266 

272 

278 

285 

291 

297 

303 

7 

4/9 

713 

309 

315 

321 

327 

333 

339 

345 

352 

358 

364 

8 

5/6 

714 

370 

376 

382 

388 

394 

400 

406 

412 

418 

425 

9 

6/3 

715 

431 

437 

443 

449 

455 

461 

467 

473 

479 

485 

716 

491 

497 

503 

509 

516 

522 

528 

534 

540 

546 

717 

552 

558 

564 

570 

576 

582 

588 

594 

600 

606 

718 

612 

618 

625 

631 

637 

643 

649 

655 

661 

667 

719 

673 

679 

685 

691 

697 

703 

709 

715 

721 

727 

720 

733 

739 

745 

751 

757 

763 

769 

775 

781 

788 

721 

794 

800 

806 

812 

818 

824 

830 

836 

842 

848 

6 

722 

854 

860 

866 

872 

878 

884 

890 

896 

902 

908 

I 
2 
3 
4 

0/6 

1/2 

1/8 

2/4 

3/6 

4/8 
5/4 

723 

914 

920 

926 

932 

938 

944 

950 

956 

962 

968 

,724 

974 

980 

986 

992 

998 

^004  *OIO 

*oi6 

*022 

*028 

725 

86034 

040 

046 

052 

058 

064 

070 

076 

082 

088 

726 

094 

100 

106 

112 

118 

124 

130 

136 

141 

147 

727 

153 

159 

16s 

171 

177 

183 

189 

195 

201 

207 

7 

728 

213 

219 

225 

231 

237 

243 

249 

255 

261 

267 

729 

273 

279 

285 

291 

297 

303 

308 

314 

320 

326 

9 

730 

332 

338 

344 

350 

356 

362 

368 

374 

380 

386 

731 

392 

398 

404 

410 

415 

421 

427 

433 

439 

445 

732 

451 

457 

463 

469 

475 

481 

487 

493 

499 

504 

733 

510 

516 

522 

528 

534 

540 

546 

552 

558 

564 

734 

570 

576 

S8i 

587 

593 

599 

605 

611 

617 

623 

735 

629 

635 

641 

646 

652 

658 

664 

670 

676 

682 

736 

688 

694 

700 

705 

711 

717 

723 

729 

735 

741 

5 

737 

747 

753 

759 

764 

770 

776 

782 

788 

794 

800 

738 

806 

812 

817 

823 

829 

835 

841 

847 

853 

859 

I 

0/5 

739 

864 

870 

876 

882 

888 

894 

900 

906 

911 

917 

2 
3 
4 

1,0 
1/5 

2/0 

2/5 

3/0 
3/5 
4/0 
4,5 

740 

923 

929 

935 

941 

947 

953 

958 

964 

970 

976 

741 

982 

988 

994 

999 

*ooS 

*oii 

*oi7  ^023  *o29  *035 

742 

87  040 

046 

052 

058 

064 

070 

075 

081 

087 

093 

7 
8 

743 

099 

loS 

III 

116 

122 

128 

134 

140 

f46 

151 

744 

157 

163 

169 

175 

181 

186 

192 

198 

204 

210 

9 

745 

216 

221 

227 

233 

239 

245 

251 

256 

262 

268 

746 

274 

280 

286 

291 

297 

303 

309 

315 

320 

326 

747 

332 

338 

344 

349 

355 

361 

367 

373 

379 

384 

748 

390 

396 

402 

408 

413 

419 

425 

431 

437 

442 

749 

448 

454 

460 

466 

471 

477 

483 

489 

495 

500 

750 

S06 

512 

518 

523 

529 

535 

541 

547 

552 

558 

N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P.P. 

N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P.P. 

750 

87506 

512 

518 

523 

529 

53S 

541 

547 

552 

558 

751 

564 

570 

576 

581 

587 

593 

599 

604 

610 

616 

752 

622 

628 

633 

639 

645 

651 

656 

662 

668 

674 

753 

679 

685 

691 

697 

703 

708 

714 

720 

726 

731 

754 

737 

743 

749 

754 

760 

766 

772 

777 

783 

789 

755 

795 

800 

806 

812 

818 

823 

829 

835 

841 

846 

756 

852 

858 

864 

869 

875 

881 

887 

892 

898 

904 

757 

910 

915 

921 

927 

933 

938 

944 

950 

955 

961 

758 

967 

973 

978 

984 

990 

996 

*ooi 

*oo7  *oi3 

*oi8 

759 

88024 

030 

036 

041 

047 

053 

058 

064 

070 

076 

760 

081 

087 

093 

098 

104 

no 

116 

121 

127 

133 

761 

138 

144 

150 

156 

161 

167 

173 

178 

184 

190 

6 

762 

19S 

201 

207 

213 

218 

224 

230 

23S 

241 

247 

0,6 

763 

252 

258 

264 

270 

275 

281 

287 

292 

298 

304 

I 

764 

309 

315 

321 

326 

332 

338 

343 

349 

355 

360 

2 

3 
4 

% 

1,2 
1,8 
2.4 
3.0 
3,6 

^'l 
4,8 

5/4 

765 

366 

372 

377 

383 

389 

395 

400 

406 

412 

417 

766 

423 

429 

434 

440 

446 

451 

457 

463 

468 

474 

767 

480 

485 

491 

497 

502 

508 

513 

519 

525 

530 

768 

536 

542 

547 

553 

559 

564 

570 

576 

581 

587 

769 

593 

598 

604 

610 

615 

621 

627 

632 

638 

643 

9 

770 

649 

65s 

660 

666 

672 

677 

683 

689 

694 

700 

771 

700 

711 

717 

722 

728 

734 

739 

745 

750 

756 

772 

762 

767 

773 

779 

784 

790 

795 

801 

807 

812 

773 

818 

824 

829 

835 

840 

846 

852 

857 

863 

868 

774 

874 

880 

885 

891 

897 

902 

908 

913 

919 

925 

775 

930 

936 

941 

947 

953 

958 

964 

969 

975 

981 

776 

986 

992 

997  »oo3  *oo9 

*oi4  *020  *o25  ^031  #037 

m 

89042 

048 

053 

059 

064 

070 

076 

081 

087 

092 

778 

098 

104 

109 

115 

120 

126 

131 

137 

143 

148 

779 
780 

154 

159 

165 

170 

176 

182 

187 

193 

198 

204 

209 

21S 

221 

226 

232 

237 

243 

248 

254 

260 

781 

265 

271 

276 

282 

28^ 

293 

298 

304 

310 

315 

5 

782 

321 

326 

332 

337 

343 

348 

354 

360 

36s 

371 

783 

376 

382 

387 

393 

398 

404 

409 

415 

421 

426 

I 

0/5 

784 

432 

437 

443 

448 

454 

459 

465 

470 

476 

481 

2 

3 
4 

1,0 

1.5 
2,0 

2,5 
3,0 
3,5 
4,0 
4/5 

785 

487 

492 

498 

504 

509 

515 

520 

526 

531 

537 

786 

542 

548 

553 

559 

564 

570 

575 

581 

586 

592 

787 

597 

603 

609 

614 

620 

625 

631 

636 

642 

647 

I 

788 

658 

664 

669 

675 

680 

686 

691 

697 

702 

789 

708 

713 

719 

724 

730 

73S 

741 

746 

752 

757 

9 

790 

763 

768 

774 

779 

785 

790 

796 

801 

807 

812 

791 

818 

823 

829 

834 

840 

84S 

851 

856 

862 

867 

792 

873 

878 

883 

889 

894 

900 

905 

9" 

916 

922 

793 

927 

933 

938 

944 

949 

955 

960 

966 

971 

977 

794 

982 

988 

993 

998 

♦004 

*oo9 

#015  *020 

*026 

*03i 

795 

90037 

042 

048 

053 

059 

064 

069 

075 

080 

086 

796 

091 

097 

102 

108 

113 

119 

124 

129 

135 

140 

797 

146 

151 

157 

162 

168 

173 

179 

184 

189 

195 

798 

200 

206 

211 

217 

222 

227 

233 

238 

244 

249 

799 

25s 

260 

266 

271 

276 

282 

287 

293 

298 

304 

800 

309 

314 

320 

325 

331 

336 

342 

347 

352 

358 

N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P.P. 

N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P.P. 

800 

90309 

314 

320 

325 

331 

336 

342 

347 

352 

358 

8oi 

363 

369 

374 

380 

385 

390 

396 

401 

407 

412 

802 

417 

423 

428 

434 

439 

445 

450 

455 

461 

466 

803 

472 

477 

482 

488 

493 

499 

504 

509 

515 

520 

804 

526 

531 

536 

542 

547 

553 

558 

563 

569 

574 

805 

580 

585 

590 

596 

601 

607 

612 

617 

623 

628 

806 

634 

639 

644 

650 

655 

660 

666 

671 

677 

682 

807 

687 

693 

698 

703 

709 

714 

720 

725 

730 

736 

808 

741 

747 

752 

757 

763 

768 

773 

779 

784 

789 

809 

795 

800 

806 

811 

816 

822 

827 

832 

838 

843 

. 

810 

849 

854 

859 

865 

870 

875 

881 

886 

891 

897 

811 

902 

907 

913 

918 

924 

929 

934 

940 

945 

950 

R 

812 

956 

961 

966 

972 

977 

982 

988 

993 

998  *oo4 

0,6 

?'3 

91009 

014 

020 

025 

030 

036 

041 

046 

052 

057 

I 

814 

062 

068 

073 

078 

084 

089 

094 

100 

105 

no 

2 

3 

4 

5 
6 

1,2 

1,8 
2.4 
3/0 
3,6 
4,2 
4,8 
5,4 

^'1 

116 

121 

126 

132 

137 

142 

148 

153 

158 

164 

816 

169 

174 

180 

185 

190 

196 

201 

206 

212 

217 

I'^l 

222 

228 

233 

238 

243 

249 

254 

259 

265 

270 

7 
8 

818 

27s 

281 

286 

291 

297 

302 

307 

312 

318 

323 

819 

328 

334 

339 

344 

350 

355 

360 

365 

371 

9 

820 

381 

387 

392 

397 

403 

408 

413 

418 

424 

429 

821 

434 

440 

445 

450 

455 

461 

466 

471 

477 

482 

822 

487 

492 

498 

503 

508 

514 

519 

524 

529 

535 

823 

540 

545 

551 

556 

561 

566 

572 

577 

582 

587 

824 

593 

598 

603 

609 

614 

619 

624 

630 

635 

640 

825 

64S 

651 

656 

661 

666 

672 

677 

682 

687 

693 

826 

698 

703 

709 

714 

719 

724 

730 

735 

740 

745 

827 

751 

756 

761 

766 

772 

777 

782 

787 

793 

798 

828 

803 

808 

814 

819 

824 

829 

834 

840 

845 

850 

829 

85s 

861 

866 

871 

876 

882 

887 

892 

897 

903 

830 

908 

913 

918 

924 

929 

934 

939 

944 

95° 

955 

831 

960 

965 

971 

976 

981 

986 

991 

997  *oo2  ^007 

5 

832 

92  012 

018 

023 

028 

033 

038 

044 

049 

054 

059 

833 

065 

070 

075 

080 

085 

091 

096 

lOI 

106 

III 

I 

o,S 

834 

117 

122 

127 

132 

137 

143 

148 

153 

158 

163 

2 

3 
4 

1,0 

1,5 
2,0 

2,5 

835 

169 

174 

179 

184 

189 

195 

200 

205 

210 

215 

836 

221 

226 

231 

236 

241 

247 

252 

257 

262 

267 

837 

273 

278 

283 

288 

293 

298 

304 

309 

314 

319 

I 

3,0 
3,5 
4,0 
4,5 

838 

324 

330 

335 

340 

345 

350 

355 

361 

366 

371 

839 

376 

381 

387 

392 

397 

402 

407 

412 

418 

423 

9 

840 

428 

433 

438 

443 

449 

454 

459 

464 

469 

474 

841 

480 

490 

495 

500 

505 

511 

516 

521 

526 

842 

531 

536 

542 

547 

552 

557 

562 

567 

572 

578 

843 

.   583 

588 

598 

609 

614 

619 

624 

629 

844 

634 

639 

645 

650 

655 

660 

665 

670 

675 

681 

84s 

686 

691 

696 

701 

706 

711 

716 

722 

727 

732 

846 

737 

742 

747 

752 

758 

763 

768 

773 

778 

783 

847 

788 

793 

799 

804 

809 

814 

819 

824 

829 

834 

848 

840 

845 

850 

855 

860 

865 

870 

875 

881 

886 

849 

891 

896 

901 

906 

911 

916 

921 

927 

932 

937 

850 

942 

947 

952 

957 

962 

967 

973 

978 

983 

988 

N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P.P. 

N. 

L.  0 

1 

2 

3 

4 

5   6 

7 

8 

9 

P.P. 

850 

92942 

947 

952 

957 

962 

967  973 

978 

983 

988 

851 

993 

998 

*oo3 

*oo8 

*oi3 

^018  #024  ^029  #034  *039 

852 

93044 

049 

054 

059 

064 

069  075 

080 

085 

090 

853 

09? 

100 

105 

no 

115 

120  125 

131 

141 

854 

146 

151 

156 

161 

166 

171  176 

181 

186 

192 

85s 

197 

202 

207 

212 

217 

222  227 

232 

237 

242 

856 

247 

252 

258 

263 

268 

273  278 

283 

288 

293 

6 

857 

298 

303 

308 

313 

318 

323  328 

334 

339 

344 

0,6 

1/2 

1/8 

2/4 

3'2 
3/6 

4/2 

4/8 
5/4 

858 

349 

354 

359 

364 

369 

374  379 

384 

389 

394 

I 

3 

4 

5 
6 

859 

399 

404 

409 

414 

420 

425  430 

435 

440 

445 

860 

450 

455 

460 

465 

470 

47S  480 

485 

490 

495 

861 

500 

50S 

510 

515 

520 

526  531 
576  581 

536 

541 

546 

862 

551 

556 

561 

566 

571 

586 

591 

596 

7 
8 

863 

601 

606 

611 

616 

621 

626  631 

636 

641 

646 

864 

651 

656 

661 

666 

671 

676  682 

687 

692 

697 

9 

865 

702 

707 

712 

717 

722 

727  732 

737 

742 

747 

866 

752 

757 

762 

767 

772 

777    782 

787 

792 

797 

867 

802 

807 

812 

817 

822 

827  832 

837 

842 

847 

868 

852 

857 

862 

867 

872 

877  882 

887 

892 

897 

869 

902 

907 

912 

917 

922 

927  932 

937 

942 

947 

870 

952 

957 

962 

967 

972 

977  982 

987 

992 

997 

871 

94002 

007 

012 

017 

022 

027  032 

037 

042 

047 

ii 

872 

052 

057 

062 

067 

072 

077  082 

086 

091 

096 

873 

lOI 

106 

III 

116 

121 

126  131 

136 

141 

146 

I 

0/5 

874 

151 

156 

161 

166 

171 

176  181 

186 

191 

196 

2 

3 
4 

i 

1,0 
1/5 

2/0 

2,5 

3/0 
3/5 
4/0 
4/5 

875 

201 

206 

211 

216 

221 

226  231 

236 

240 

245 

876 

250 

255 

260 

265 

270 

275  280 

28S 

290 

295 

877 

300 

305 

310 

315 

320 

325  330 

335 

340 

345 

7 

8 

878 

349 

354 

359 

364 

369 

374  379 

384 

389 

394 

879 

399 

404 

409 

414 

419 

424  429 

433 

438 

443 

9 

880 

448 

453 

458 

463 

468 

473  478 

483 

488 

493 

881 

498 

503 

507 

512 

517 

522  527 

532 

537 

542 

882 

547 

552 

557 

•362 

567 

571  ,  576 

581 

586 

591 

883 

596 

601 

606 

^i^^ 

^616 

621'  626 

630 

635 

640 

884 

645 

650 

655 

660*^  665 

670  675 

680 

685 

689 

88s 

694 

699 

704 

709 

714 

719  724 

729 

734 

738 

886 

743 

748 

753 

758 

763 

768  773 

778 

783 

787 

d 

887 

792 

797 

802 

807 

812 

817  822 

827 

832 

836 

888 

&41 

846 

851 

856 

861 

866  871 

876 

880 

88s 

I 

°4 

1,2 

1/6 

2/0 

"A 

889 

890 

895 

900 

905 

910 

915  919 

924 

929 

934 

2 
3 
4 
5 
6 

I 

9 

890 

939 

944 

949 

954 

959 

963  968 

973 

978 

983 

891 

993 

998  *002  ^OOJ 

#012  ^017  *022  *027  ^032 

892 

95036 

041 

046 

051 

056 

061   066 

071 

075 

080 

893 

085 

090 

095 

100 

105 

109   114 

119 

124 

129 

i% 

894 

134 

139 

143 

148 

153 

158  i6a 

168 

173 

177 

895 

182 

187 

192 

197 

202 

207   211 

216 

221 

226 

896 

231 

236 

240 

245 

250 

255   260 

265 

270 

274 

897 

279 

284 

289 

294 

299 

303   308 

313 

3^^ 

323 

898 

328 

332 

337 

342 

347 

352  357 

361 

366 

371 

899* 

376 

381 

386 

390 

395 

400  405 

410 

415 

419 

900 

424 

429 

434 

439 

444 

448  453 

458 

463 

468 

N. 

L.  0 

1 

2 

3 

4 

5   6 

7 

8 

9 

P.P. 

N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P.P. 

900 

95424 

429 

434 

439 

444 

448 

453 

458 

463 

468 

901 

472 

477 

482 

487 

492 

497 

501 

506 

511 

516 

902 

521 

52S 

530 

535 

540 

545 

550 

554 

559 

564 

903 

569 

574 

578 

583 

588 

593 

598 

612 

904 

617 

626 

631 

636 

641 

646 

650 

655 

660 

90s 

665 

670 

674 

679 

684 

689 

694 

698 

703 

708 

906 

713 

718 

722 

727 

732 

737 

742 

746 

751 

756 

907 

761 

766 

770 

775 

780 

785 

789 

794 

799 

804 

908 

809 

813 

818 

823 

828 

832 

837 

842 

847 

852 

909 

856 

861 

866 

871 

875 

880 

885 

890 

895 

899 

910 

904 

909 

914 

918 

923 

928 

933 

938 

942 

947 

911 

952 

957 

961 

966 

971 

976 

980 

985 

990 

995 

5 

912 

999  *oo4  ^009  *oi4  *oi9 

*023 

*028 

*o33 

*038 

*042 

I 

o.S 
1,0 

913 

96047 

052 

057 

061 

066 

071 

076 

080 

085 

090 

2 

914 

095 

099 

104 

109 

114 

118 

123 

128 

133 

137 

3 
4 

1.5 
2,0 

2,5 

915 

142 

147 

152 

156 

161 

166 

171 

175 

180 

185 

916 

190 

194 

199 

204 

209 

213 

218 

223 

227 

232 

6 

3/0 

917 

237 

242 

246 

251 

256 

261 

265 

270 

275 

280 

7 

3/5 

918 

284 

289 

294 

298 

303 

308 

313 

317 

322 

327 

8 

4/0 

919 

332 

336 

341 

346 

3So 

355 

360 

365 

369 

374 

9 

4.5 

920 

379 

384 

388 

393 

398 

402 

407 

412 

417 

421 

921 

426 

431 

435 

440 

445 

450 

454 

459 

464 

468 

922 

473 

478 

483 

487 

492 

497 

501 

506 

5" 

515 

923 

520 

525 

530 

534 

539 

544 

548 

553 

558 

562 

924 

567 

572 

577 

581 

586 

591 

595 

600 

605 

609 

925 

614 

619 

624 

628 

633 

638 

642 

647 

652 

656 

926 

661 

666 

670 

675 

680 

685 

689 

694 

699 

703 

927 

708 

713 

717 

722 

727 

731 

736 

741 

745 

750 

928 

755 

759 

764 

769 

774 

778 

783 

788 

792 

797 

929 

802 

806 

811 

816 

820 

825 

830 

834 

839 

844 

930 

848 

^53 

858 

862 

867 

872 

876 

881 

886 

890 

931 

895 

900 

904 

909 

914 

918 

923 

928 

932 

937 

4 

932 

^il 

946 

951 

956 

960 

965 

970 

97^ 

979 

984 

I 

1/2 

1/6 

2  0 

933 

988 

993 

997  *002  *007 

*OII 

*oi6 

^021  ^025  ^030 

934 

97035 

039 

044 

049 

053 

058 

063 

067 

072 

077 

2 
3 
4 

935 

081 

086 

090 

095 

100 

104 

109 

114 

118 

123 

936 

128 

132 

137 

146 

151 

155 

160 

165 

169 

2/4 
2  8 

937 

174 

179 

183 

188 

192 

197 

202 

206 

211 

216 

7 

938 

220 

225 

230 

234 

239 

243 

248 

253 

257 

262 

te 

939 

267 

271 

276 

280 

285 

290 

294 

299 

304 

308 

9 

940 

313 

317 

322 

327 

331 

336 

340 

345 

350 

354 

941 

359 

364 

368 

373 

377 

382 

387 

391 

396 

400 

942 

405 

410 

414 

419 

424 

428 

433 

437 

442 

447 

943 

451 

456 

460 

465 

470 

474 

479 

483 

488 

493 

944 

497 

502 

506 

511 

516 

520 

525 

529 

534 

539 

945 

543 

548 

552 

557 

562 

566 

571 

575 

580 

585 

946 

589 

594 

598 

603 

607 

612 

617 

621 

626 

630 

947 

635 

640 

644 

649 

653 

658 

663 

667 

672 

676 

948 

681 

685 

690 

695 

699 

704 

708 

713 

717 

722 

949 

727 

731 

736 

740 

745 

749 

754 

759 

763 

768 

950 

772 

777 

782 

786 

791 

795 

800 

804 

809 

813 

N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P.P. 

N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P.P. 

950 

97772 

in 

782 

786 

791 

795 

800 

804 

809 

813 

951 

818 

823 

827 

832 

836 

841 

845 

850 

855 

859 

952 

864 

868 

873 

877 

882 

886 

891 

896 

900 

905 

953 

909 

914 

918 

923 

928 

932 

937 

941 

946 

950 

954 

955 

959 

964 

968 

973 

978 

982 

987 

991 

996 

955 

98  000 

005 

009 

014 

019 

023 

028 

032 

037 

041 

956 

046 

050 

055 

059 

064 

068 

073 

078 

082 

087 

957 

091 

096 

100 

105 

109 

114 

118 

123 

127 

132 

958 

137 

141 

146 

ISO 

155 

159 

164 

168 

173 

177 

959 

182 

186 

191 

195 

200 

204 

209 

214 

218 

223 

960 

227 

232 

236 

241 

245 

250 

254 

259 

263 

268 

961 

272 

277 

281 

286 

290 

295 

299 

304 

308 

313 

5 

962 

318 

322 

327 

331 

336 

340 

345 

349 

354 

358 

I 

0,5 
I  0 

963 

363 

367 

372 

376 

381 

385 

390 

394 

399 

403 

2 

964 

408 

412 

417 

421 

426 

430 

435 

439 

444 

448 

3 

4 
5 

lis 

2,0 

2,5 

9^1 

453 

457 

462 

466 

471 

475 

480 

484 

489 

493 

966 

498 

502 

507 

5^^ 

516 

520 

525 

529 

534 

538 

6 

3/0 

967 

543 

547 

552 

556 

561 

565 

570 

574 

579 

583 

7 

3/5 

968 

588 

592 

597 

601 

605 

610 

614 

619 

623 

628 

8 

4/0 

969 

632 

637 

641 

646 

650 

655 

659 

664 

668 

673 

9 

4,5 

970 

677 

682 

686 

691 

695 

700 

704 

709 

713 

717 

971 

722 

726 

731 

735 

740 

744 

749 

758 

762 

972 

767 

771 

776 

780 

784 

789 

793 

798 

802 

807 

973 

811 

816 

820 

825 

829 

834 

838 

843 

847 

851 

974 

856 

860 

865 

869 

874 

878 

883 

887 

892 

896 

975 

900 

905 

909 

914 

918 

923 

927 

932 

936 

941 

976 

945 

949 

954 

958 

963 

967 

972 

976 

981 

985 

977 

989 

994 

998  *oo3 

*oo7 

*OI2 

*oi6 

*02I 

*025  *029 

978 

99  034 

038 

043 

047 

052 

056 

061 

065 

069 

074 

979 

078 

083 

087 

092 

096 

100 

105 

109 

114 

118 

980 

123 

127 

131 

136 

140 

145 

149 

154 

158 

162 

981 

167 

171 

176 

180 

185 

189 

193 

198 

202 

207 

4 

982 

211 

216 

220 

224 

229 

233 

238 

242 

247 

251 

983 

•255 

260 

264 

269 

273 

277 

282 

286 

291 

295 

I 

o;t 

1/2 

2  0 

984 

300 

304 

308 

313 

317 

322 

326 

330 

335 

339 

2 
3 
4 

9^1 

344 

348 

352 

357 

361 

366 

370 

374 

379 

383 

986 

388 

392 

396 

401 

405 

410 

414 

419 

423 

427 

2^4 
2/8 

987 

432 

436 

441 

445 

449 

454 

458 

463 

467 

471 

7 
8 

988 

476 

480 

484 

489 

493 

498 

502 

506 

511 

515 

3'? 

3/6 

989 

520 

524 

528 

533 

537 

542 

546 

550 

555 

559 

9 

990 

564 

568 

572 

577 

581 

585 

590 

594 

599 

603 

991 

607 

612 

616 

621 

625 

629 

634 

638 

642 

647 

992 

651 

656 

660 

664 

669 

673 

677 

682 

686 

691 

993 

695 

699 

704 

708 

712 

717 

721 

726 

730 

734 

994 

739 

743 

747 

752 

756 

760 

765 

769 

774 

778 

995 

782 

787 

791 

795 

800 

804 

808 

813 

817 

822 

996 

826 

830 

835 

839 

843 

848 

852 

856 

861 

865 

997 

870 

874 

878 

883 

887 

891 

896 

900 

904 

909 

998 

913 

917 

922 

926 

930 

935 

939 

944 

948 

952 

999 

957 

961 

965 

970 

974 

978 

983 

987 

991 

996 

1000 

00  000 

004 

009 

013 

017 

022 

026 

030 

035 

039 

N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P.P. 

NOTES  ON  TABLES  I  AND  II. 

The  logarithms  of  numbers  are  in  general  incommensurable. 
In  these  tables  they  are  given  correct  to  five  places  of  decimals. 
If  the  sixth  place  is  5  or  more,  the  next  larger  number  is  used 
in  the  fifth  place.  Thus  log  8102  =  3.908549+;  in  five-place 
tables  this  is  written  3.90855,  the  dash  above  the  5  showing 
that  the  logarithm  is  less  than  given. 

So  log  8133  =  3.910251-;  in  five-place  tables  this  is  written 
3.91025,  the  dot  above  the  5  showing  that  the  logarithm  is  more 
than  given. 

In  the  natural  functions  of  the  angles  (Table  II)  all  numbers 
are  decimals  for  sine  and  cosine  (why  ?),  and  for  tangent  and 
cotangent,  except  where  the  decimal  point  is  used  to  indicate 
that  part  of  the  number  is  integral.  When  no  decimal  point 
is  printed  in  the  tables  it  is  to  be  understood.  When  the 
natural  function  is  a  pure  decimal  the  characteristic  of  the 
logarithm  is  negative.  Accordingly,  in  the  tables  10  is  added, 
and  in  the  result  this  must  be  allowed  for.     Thus 

nat.  sin  44°  20'  =  0.69883,   log  sin  44°  20'  =  1.84437, 

or,  as  printed  in  the  tables,  9.84437,  which  means  9.84437  — ?0. 


TABLE  n. 

THE  LOGARITHMIC  AND  NATURAL  SINES,  COSESTES, 

TANGENTS,  AND  COTANGENTS  OF  ANGLES 

FROM  0°  TO  90°. 


»5 


Nat.  Sin  Log.      d. 


Nat.CoSLog.  Nat.TanLog 


Log.CotNat. 


ooooo 
029 

058 
087 
116 


646373 

6.76476 
6.94085 
7-06579 


0014s 

175 

204 

233 

262 


7.10270 
7.24188 
7.30882 
7.36682 
7.41797 


00291 
320 

349 
378 
407 


746373 
7-50512 
7.54291 
7-57767 
7-60985 


00436 
465 
495 

524 
553 


7.63982 
7.66784 
7.69417 
7.71900 
7.74248 


00582 
611 
640 
669 
698 


7-76475 
7.78594 
7.8061$ 

7-82545 
7-8439^ 


00727 
756 
785 
814 

844 

00873 

902 

931 
960 


7.861O0 
7.87870 
7.89509 
7.91088 
7.92612 


7.94084 
7-95508 
7.96887 
7.98223 
7.99520 


01018 
047 
076 
105 
134 


8.00779 
8.02002 
8.03192 
8.04350 
8.ot;4'78 


01 164 

193 

222 

251 

280 


8.0J57i 
8.07650 
8.0S696 
8.09718 
8.10717 


01309 
338 
367 
396 
425 


8.1 1693 
8.12647 
8.13581 
8.14495 
8-15391 


01454 
483 
513 
542 
571 


8.16268 
8.17128 
8.17971 
8.18798 
8.19610 


01600 
629 
658 
687 
716 
745 


8.20407 
8.21 189 
8.21958 
8.22713 
8.23456 
8.24186 


30103 
17609 
12494 
9691 
7918 
6694 
5800 
51 15 
4576 
4139 
3779 
3476 
3218 
2997 
2802 
2633 
2483 
2348 
2227 
2 1 19 
2021 
1930 
1848 

1773 
1704 
1639 
1579 
1524 
1472 
1424 
1379 
1336 
1297 

1259 
1223 
1 190 
1158 
1 128 

IIOO 

1072 

1046 

1022 

999 

976 

954 
934 
914 
896 
877 
860 

843 
827 
812 

797 
782 
769 
755 
743 
730 


loooo  0.00000 
000  0.00000 
000  0.00000 
000  0.00000 
000  0.00000 


loooo  0.00000 
000  0.00000 
000  0.00000 
000  0.00000 
000  0.00000 


loooo  0.00000 

99999  0.00000 

999  0.00000 

999  0.00000 

999  0.00000 


99999  0.00000 

999  0.00000 

999  9-99999 

999  9-99999 

998  9-99999 


99998  9.99999 
998  9-99999 
998  9-99999 
998  9-99999 
998  9-99999 


99997 
997 
997 
997 
996 


9.99999 
9.99999 
9.99999 
9.99999 
9-99998 


99996 
996 
996 
995 
995 


9.99998 
9.99998 
9.99998 
9.99998 
9-99998 


99995 
995 
994 
994 
994 


9.99998 
9.99998 
9.99997 

9-99997 
9.99997 


99993 
993 
993 
992 
992 


9-99997 
9-99997 
9.99997 
9.99997 
9-99996 


99991 
991 
991 
990 
990 


9-99996 
9.99996 
9-99996 
9-99996 
9-99996 


99989 
989 
989 


9-99995 
9-9999$ 
9-99995 
9-99995 
9-99995 


99987  9-99994 
987  9-99994 
986  9-99994 
986  9.99994 
985  9.99994 
985  9.99993 


029 
058 
087 
116 


6.46373 
6.76476 
6.94085 
7-06579 


0014s 

175 
204 

233 

262 


7.16270 
7.24188 
7.30882 
7.36682 
7.41797 


00291 
320 
349 
378 
407 


7.46373 
7.50512 
7.54291 

7-57767 
7.60986 


00436 
46s 
495 
524 

553 


7-63982 
7.66785 
7.69418 
7.71900 
7.74248 


00582 
611 
640 
669 


7.76476 
7-7859$ 
7.80615 
7-82546 
7-84394 


00727 
756 
785 
815 
844 


7.86167 
7.87871 
7.89510 
7.91089 
7-92613 


00873 
902 

931 
960 


7.94086 
7-95510 
7.96889 
7-98225 
7-99522 


047 
076 
105 
135 


8.00781 
8.02004 
8.03194 

8.04353 
8.05481 


01 164 

193 
222 

251 
280 


8.06581 
8.07653 
8.08700 
8.09722 
8.10720 


01309 
338 
367 
396 
425 


8.11696 
8.12651 

8.13585 
8.14500 

8-15395 


01455 
484 

513 
542 
571 


8.16273 

8-17133 
8.17976 
8.18804 
8.19616 


01600 
629 
658 
687 
716 
746 


8.20413 
8.21 195 
8.21964 
8.22720 
8.23462 
8.24192 


30103 
17609 
12494 
9691 
7918 
6694 
5800 
5115 
4576 
4139 
3779 
3476 
3219 
2996 
2803 
2633 
2482 
2348 
2228 
21 19 
2020 

193 1 
1848 

1773 
1704 
1639 
1579 
1524 
1473 
1424 
1379 
1336 
1297 

1259 
1223 
1 190 

1159 
1128 

IIOO 

1072 
1047 
1022 
998 
976 

955 
934 
915 
895 
878 
860 

843 
828 
812 

797 
782 
769 
756 
742 
730 


3-53627 
3.23524 
3-05915 
2.93421 


3437-7 
171B.9 

1145-9 
859.44 


2.83730 
2.75812 
2.691 18 
2.63318 
2.58203 


687.55 
572.96 
491.11 
429.72 
381.97 


2.53627 
2.49488 
2.45709 
2.42233 
2.39014 


2.36018 

2.33215 
2.30582 
2.28100 
2.25752 


343-77 
312.52 
286.48 
264.44 
^5:55 
229.18 
214.86 
202.22 
190.98 
180.93 


2.23524 
2.21405 
2.19385 
2.17454 
2.15606 


171.89 
163.70 
156.26 
149.47 
143.24 


2.13833 
2.12129 
2.10490 
2.0891 1 
2.07387 


137-51 
132.22 
127.32 
122.77 
118.54 


2.05914 
2.04490 
2.03111 

2.01775 
2.00478 


114-59 
110,89 

107.43 
104.17 

lOI.II 


1.99219 
1.97996 
1.96806 
1.95647 
1.94519 


1. 93419 
1.92347 
1. 91300 
1.90278 
1.89280 


98.218 
95.489 
92.908 
90.463 
88.144 
85.940 
83.844 
81.847 

79-943 
78.126 


1.88304 
1.87349 

1.86415 
1.85500 
1.84605 


76.390 
74.729 

73-139 

71.615 

70.153 


1.83727 
1.82867 
1.82024 
1.81196 
1.80384 


68.750 
67.402 
66.105 
64.858 
63-657 


1.79587 
1.78805 
1.78036 
1.77280 
1-76538 
1.75808 


62.499 

61.383 
60.306 
59.266 
58.261 
57-290 


Nat.  Cos  Log.      d. 


Nat.  Sin  Log 


Nat.CotLog 


89' 


d.    Log.TanNat. 


>i^ 


Nat.  Sin  Log.    d. 


Nat.  Cos  Log 


Nat  .Tan  Log. 


c.d. 


Log.  Cot  Nat, 


01745 
774 
803 
832 
862 


8.24186 
8.24903 
8.25609 
8.26304 
8.26988 


01891 
920 

949 

978 

02007 


8.27661 
8.28324 
8.28977 
8.29621 
8.30255 


02036 
065 
094 
123 
152 


8.30879 

8.31495 
8.32103 
8.32702 
8.33292 


02181 
211 
240 
269 
298 


8-33875 
8.34450 
8.35018 

8.35578 
8.36131 


02327 
356 
385 
414 

443 


8.36678 
8.37217 
8.37750 
8.38276 
8.38796 


25 

26 
27 

28 

_?9 
30 
31 
32 
33 
_31 
35 
36 
37 
38 
39 
40 

41 
42 

43 
44_ 
45 

46 

47 
48 

49_ 
50 

SI 
52 
53 
54 


02472 
501 
530 
560 

589 


8.39310 
8.39818 
840320 
840816 
841307 


02618 
647 
676 
705 
734 


8.41792 
8.42272 
8.42746 
843216 
8.43680 


02763 
792 
821 
850 
879 


8.44139 

8.44594 
845044 
845489 
8.45930 


02908 
938 
967 
996 

03025 


846366 
8.46799 
8.47226 
8.47650 
8.48069 


03054 
083 
112 
141 
170 


8.48485 
848896 

849304 
849708 
8.50108 


03199 
228 

257 
286 
316 


8.50504 
8.50897 
8.51287 
8.51673 
8.52055 


03345 
374 
403 
432 
461 
490 


8.52434 
8.52810 
8.53183 
8.53552 
8.53919 
8.54282 


717 
706 

695 
684 

673 
663 

653 
644 

634 
624 
616 
608 
599 
590 
583 

575 
568 
560 
553 
547 
539 
533 
526 
520 

514 
508 
502 
496 
491 

485 
480 
474 
470 
464 

459 
455 
450 

445 
441 

436 

433 
427 

424 
419 
416 
411 
408 

404 
400 

396 
393 
390 
386 
382 

379 
376 
373 
369 
367 
363 


99985 
984 
984 
983 
983 


9.99993 
9-99993 
999993 
9-99993 
9-99992 


01746 

775 
804 

833 
862 


99982 
982 
981 
980 
980 


9.99992 
9.99992 
9-99992 
9-99992 
9.99991 


01891 
920 
949 
978 

02007 


99979 
979 
978 

977 
977 


9.99991 
9.99991 
9.99990 
9-99990 
9-99990 


02036 
066 

095 
124 

153 


99976 
976 
975 
974 
974 


9-99990 
9-99989 
9-99989 
9.99989 

9-99989 


02182 
211 
240 

269 


99973 
972 
972 
971 
970 


9.99988 
9.99988 
9.99988 
9.99987 
9-99987 


02328 

357 
386 

415 
444 


8.24192 
8.24910 
8.25616 
8.26312 
8.26996 
8.27669 
8.28332 
8.28986 
8.29629 
8.30263 
8.30888 

8.31505 
8.321 12 

8.327x1 
8.33302 
8.33886 
8.34461 
8.35029 
8-35590 
8.36143 
8.36689 
8.37229 
8.37762 
8.38289 
8.38809 


99969 
969 
968 
967 
966 


9-99987 
9.99986 
9.99986 
9.99986 
9.99985 


02473 
502 

531 
560 

589 


99966 

965 
964 
963 
963 


9.99985 
9-99985 
9-99984 
9-99984 
9.99984 


02619 
648 
677 
706 
735 


99962 
961 
960 
959 
959 


9-99983 
9-99983 
9.99983 
9-99982 
9.99982 


02764 

793 
822 

851 
881 


8-39323 
8.39832 
840334 
840830 

841321 
8.41807 
8.42287 
842762 
8.43232 
8.43696 
844156 


99958 
957 
956 
955 
954 


9.99982 
9.99981 
9.99981 
9.99981 
9.99980 


02910 

939 
968 

997 
03026 


99953 
952 
952 
951 
950 


9.99980 
9.99979 
9.99979 
9.99979 
9-99978 


03055 
084 
114 

143 
172 


8.4461 1 
8.45061 
8.45507 
845948 

846385 
846817 

8-47245    424 

847669  12 

84808^14- 
flf5?5    412 


99949 
948 
947 
946 

945 


9-99978 
9-99977 
9-99977 
9-99977 
9-99976 


03201 
230 

259 
288 

317 


99944 
943 
942 
941 
940 
939 


9.99976 
9-99975 
9-99975 
9-99974 
9-99974 
9-99974 


03346 
376 
405 
434 
463 
492 


8.48917 
8.49325 
8.49729 
8.50130 

8.50527 
8.50920 
8.51310 
8.51696 
8.52079 
8.52459 
8.52835 
8.53208 
8.53578 
8.53945 
8.54308 


718 
706 
696 
684 

673 
663 
654 
643 
634 
625 
617 
607 
599 
591 
584 

575 
568 

561 
553 
546 
540 
533 
527 
520 

514 
509 
502 
496 
491 
486 
480 

475 
470 
464 
460 

455 
450 
446 
441 

437 
432 
428 
424 


408 

404 
401 

397 
393 
390 
386 
383 
380 
376 
373 
370 
367 
363 


.75808 
•75090 
•74384 
•73688 
.73004 


57.290 
56.351 
55-442 
54-561 
53-709 


.72331 
.71668 
.71014 
•70371 
•69737 


52.882 
.081 
51-303 
50.549 
49.816 


.69112 
.68495 
.67888 
.67289 
.66698 


49.104 
48.412 
47-740 
-085 
46.449 


.66114 

-65539 
.64971 
.64410 
.63857 


45.829 
.226 

44-639 

.066 

43-508 


63311 
,62771 
,62238 
,61711 
,61191 


42.964 

-433 
41.916 

411 
40.917 


60677 
60168 
59666 
59170 
58679 


40.436 

39.965 

.506 

.057 
38.618 


58193 
57713 
57238 
56768 
56304 


38.188 
37.769 

.358 
36.956 

.563 


55844 
55389 
54939 
54493 
54052 


36.178 
35.801 

431 
.070 

34.715 


53615 
53183 
52755 
52331 
519" 


34.368 
.027 

33.694 
.366 
.045 


•51495 
.51083 

.50675 
.50271 
.49870 


32.730 
.421 
.118 

31.821 
.528 


49473 
49080 
48690 
48304 
•47921 


31.242 

30.960 

.683 

412 

.145 


47541 
47165 
46792 
46422 
46055 
45692 


29.882 
.624 
•371 

.122 

28.877 

.636 


Nat.  Cos  Log.    d.    Nat.  Sin  Log.  Nat.  Cot  Log.  c.d.  Log.  Tan  Nat.    ' 

88° 


f 

Nat.  S 

in  Log.  d. 

Nat.  Cos  Log. 

Nat.Tan  Log. 

c.d. 

Log.  Cot  Nat. 

r" 

0 

03490 

8.54282 

360 
357 
355 
351 
349 
346 
343 

99939 

9-99974 

03492 

8.54308 

361 
358 
355 

145692 

28.636 

60 

I 

519 

8.54642 

938 

9.99973 

521 

8.54669 

I-4533I 

.399 

59 

2 

54a 

8.54999 

937 

9-99973 

550 

8.55027 

1.44973 

.166 

58 

3 

577 

8-55354 

936 

9.99972 

579 

8.55382 

1.44618 

27.937 

57 

4 

606 

8-55705 

935 

9.99972 

609 

8.55734 

352 
349 
346 
344 

1.44266 

.712 

56 

5 

03635 

8.56054 

99934 

9.99971 

03638 

8.56083 

143917 

27.490 

55 

b 

bb4 

8.56400 

933 

9.99971 

bbj 

8.56429 

1.43571 

.271 

54 

7 

693 

8.56743 

932 

9.99970 

696 

8.56773 

1.43227 

.057 

53 

8 

723 

8.57084 

341 
337 
336 

931 

9.99970 

725 

8.57114 

34  i 
338 
336 

1.42886 

26.845 

52 

9 

752 

8.57421 

930 

9.99969 

754 

8.57452 

1.42548 

.637 

51 

10 

03781 

8-57757 

99929 

9.99969 

03783 

8.57788 

1.42212 

26.432 

50 

II 

810 

8.58089 

332 

927 

9.99968 

812 

8.58121 

333 

1.41879 

.230 

49 

12 

839 

8.58419 

330 
328 
325 
323 

926 

9.99968 

842 

8.58451 

330 
328 
326 
323 

1.41549 

.031 

48 

13 

8b8 

8.58747 

925 

9.99967 

871 

8.58779 

1.41221 

25.835 

47 

H 

897 

8.59072 

924 

9.99967 

900 

8.59105 

1.40895 

.642 

46 

15 

03926 

8.5939$ 

99923 

9-99967 

03929 

8.59428 

1.40572 

25.452 

45 

lb 

955 

8.59715 

320 

922 

9.99966 

958 

8.59749 

321 

1.40251 

.264 

44 

17 

984 

8.60033 

i-i-o 

921 

9.99966 

987 

8.60068 

319 
316 

1-39932 

.080 

43 

l8 

04013 

8.60349 

3^0 

919 

9-99965 

04016 

8.60384 

1.39616 

24.898 

42 

19 

042 

8.60662 

313 
311 

918 

9-99964 

046 

8.60698 

314 

311 
310 

307 
305 
303 
301 
299 
297 

295 
292 
291 
289 
287 
285 
284 
281 
280 

1.39302 

.719 

41 

20 

04071 

8.60973 

99917 

9-99964 

04075 

8.61009 

1.38991 

24.542 

40 

21 

100 

8.61282 

916 

9-99963 

104 

8.61319 

1.38681 

.368 

39 

22 

129 

8.61589 

307 
305 

915 

9.99963 

133 

8.61626 

1-38374 

.196 

38 

23 

159 

8.61894 

913 

9.99962 

162 

8.61931 

1.38069 

.026 

37 

24 

25 

188 

8.62196 

302 
301 
298 
296 

912 

9.99962 

191 

8.62234 

1.37766 

23.859 

36 

04217 

8.62497 

999" 

9.99961 

04220 

8.62535 

1.37465 

23-695 

35 

2b 

24b 

8.62795 

910 

9.99961 

250 

8.62834 

1.37166 

.532 

34 

27 

275 

8.63091 

909 

9.99960 

279 

8.63131 

1.36869 

.372 

33 

28 

304 

8.63385 

294 

907 

9.99960 

308 

8.63426 

.214 

32 

29 

,333 

8.63678 

293 
290 
288 

906 

9-99959 

337 

8.63718 

1.36282 

.058 

31 
30 

30 

04362 

8.63968 

99905 

9-99959 

04366 

8.64009 

I-3.599I 

22.904 

31 

391 

8.64256 

287 
284 
283 
281 

904 

9.99958 

395 

S.64298 

1-35702 

.752 

29 

32 

420 

t^^ 

902 

9-99958 

424 

8.64585 

1.35415 

.602 

28 

33 

449 

901 

9-99957 

454 

8.64870 

1-35130 

.454 

27 

34 

47B 

8.65110 

900 

9.99956 

483 

8.65154 

1.34846 

.308 

2b 

25 

35 

04507 

8.65391 

99898 

9.99956 

04512 

8.65435 

1-3456$ 

22.164 

3^ 

536 

8.65670 

897 

9.99955 

541 

8.65715 

278 
276 
274 

273 
271 
269 
268 

1.34285 

.022 

24 

565 

8.65947 

276 

896 

9-99955 

570 

1.34007 

21.881 

23 

38 

594 

8.66223 

894 

9-99954 

599 

8.66269 

1.33731 

.743 

22 

39 

623 

8.66497 

272 

893 

9-99954 

628 

8.66543 

1.33457 

.606 

21 
20 

40 

04653 

8.66769 

99892 

9-99953 

04658 

8.66816 

1-33184 

21.470 

41 

682 

8.67039 

270 
269 

267 
266 

890 

9-99952 

687 

8.67087 

1.32913 

.337 

19 

42 

711 

8.67308 

889 

9.99952 

71b 

8.67356 

1.32644 

.205 

18 

43 

740 

8.67575 

888 

9-99951 

745 

8.67624 

266 

1.32376 

.075 

17 

44 
45 

769 

8.67841 

263 

2bO 

886 

9-99951 

774 

8.67890 

264 
263 
261 

1.32110 

20.946 

lb 

04798 

8.68104 

99885 

9.99950 

04803 

8.68154 

1.31846 

20.819 

15 

4b 

827 

8.68367 

883 

9.99949 

833 

8.68417 

1. 31583 

.693 

14 

47 

856 

8.68627 

882 

9-99949 

862 

8.68678 

260 

1.31322 

.5^9 

13 

48 

885 

8.68886 

259 

258 
256 

881 

9-99948 

891 

8.68938 

258 
257 

255 
254 
252 

251 
249 
248 
246 

245 
244 

243 

1.31062 

.446 

12 

49 
50 

914 

8.69144 

879 

9.99948 

920 

8.69196 

1.30804 

•325 

II 

04943 

8.69400 

99878 

9-99947 

04949 

8.69453 

1.30547 

20.206 

10 

51 

972 

8.69654 

254 

876 

9-99946 

978 

8.69708 

1.30292 

.087 

9 

.S2 

05001 

8.69907 

253 

875 

9-99946 

05007 

8.69962 

1.30038 

19.970 

8 

53 

030 

8.70159 

252 

250 
249 

873 

9-99945 

037 

8.70214 

1.29786 

.855 

7 

54 

059 

8.70409 

872 

9.99944 

066 

8.70465 

1-29535 

.740 

b 

55 

05088 

8.70658 

99870 

9.99944 

05095 

8.70714 

1.29286 

19.627 

5 

S^ 

117 

8.70905 

247 

246 

869 

9-99943 

124 

8.70962 

1.29038 

,  .51^ 

4 

57 

146 

8.71151 

867 

9.99942 

153 

8.71208 

1.28792 

.405 

3 

5a 

175 

8.71395 

866 

9.99942 

182 

8.71453 

1.28547 

.29b 

2 

ro 

205 

8.71638 

243 

864 

9.99941 

212 

8.71697 

1.28303 

.188 

I 

234 

8.71880 

863 

9.99940 

241 

8.71940 

1.28060 

.081 

0 

Nat.  Cos  Log.  d. 

Nat.  S 

in  Log. 

Nat.  Cot  Log. 

c.d. 

Log. Tan  Nat. 

f 

87° 


3° 


Nat.  Sin  Log.    d. 


Nat.  Cos  Log, 


Nat.TanLog.  c.d. 


Log.  Cot  Nat, 


05234 
263 
292 
321 
35° 


8.71880 
8.72120 

8-72359 
8.72597 
8.72834 


05379 
408 

437 
466 

495 


8.73069 
8-73303 
8-73535 
8.73767 

8.73997 


05524 
553 
582 
611 
640 


8.74226 

8-74454 
8.74680 
8.74906 
8-75130 


05669 
698 
727 
756 
785 


8-75353 
8.75575 
8.75795 
8.76015 
8.76234 


05814 
844 
873 
902 

931 


8.76451 
8.76667 
8.76883 
8.77097 
8.77310 


05960 

989 

06018 

047 

076 


8.77522 
8.77733 
8.77943 
8.78152 
8.78360 


06105 

134 
163 
192 
221 


8.78568 
8.78774 
8.78979 
8.79183 
8.79386 


06250 
279 
308 

337 
366 


8.79588 
8.79789 
8.79990 
8.80189 
8.80388 


06395 

424 

453 

482 

5" 


8.80585 
8.80782 
8.80978 
8.81 173 
8.81367 


06540 
569 
598 
627 
656 


8.81560 
8.81752 
8.81944 
8.82134 
8.82324 


06685 
714 
743 
773 
802 


8.82513 
8.82701 
8.82888 
8.83075 
8.83261 


06831 
860 
889 
918 

947 
976 


8.83446 
8.83630 
8.83813 
8.83996 
8.84177 
8.84358 


240 

239 
238 

237 
235 
234 
232 
232 
230 
229 
228 
226 
226 
224 
223 
222 
220 
220 
219 
217 
216 
216 
214 
213 
212 
211 
210 
209 
208 
208 
206 
205 
204 
203 
202 
201 
201 
99 
99 
97 
97 
96 

95 
94 
93 
92 
92 
90 
90 
89 
88 
87 
87 
86 
85 
84 
83 
83 
81 
81 


99863 
861 
860 
858 
857 


9.99940 
9-99940 
9-99939 
9.99938 
9.99938 


05341 
270 

299 

328 

357 


241 
239 
239 


99855 
854 
852 
851 
849 


9-99937 
9.99936 
9.99936 
9-99935 
9-99934 


05387 
416 

445 
474 
503 


99847 
846 

844 
842 
841 


9-99934 
9-99933 
9.99932 
9.99932 
9-99931 


05533 

562 

591 
620 
649 


99839 
838 
836 
834 
833 


9-99930 
9.99929 
9.99929 
9.99928 
9-99927 


05678 
708 

737 
766 

795 


99831 

9.99926 

829 

9.99926 

827 

9.99925 

826 

9.99924 

824 

9-99923 

99822 

9.99923 

821 

9-99922 

819 

9.99921 

817 

9.99920 

815 

9-99920 

99813 

9.99919 

812 

9.99918 

810 

9.99917 

808 

9.99917 

806 

9.99916 

99804 

9-99915 

803 

9.99914 

801 

9-99913 

799 

9-99913 

797 

9.99912 

99795 

9-999" 

793 

9.99910 

792 

9-99909 

790 

9-99909 

788 

9.99908 

05824 

854 
883 

912 

941 


05970 

999 

06029 

058 

087 


061 16 
145 
175 
204 

233 


06262 
291 
321 
350 
379 


06408 
438 
467 
496 
525 


8.71940 

8.72181 
8.72420 
8.72659  2^9 
8.72896  '  ^37 
-^ — ^  236 
8.73132  20. 
8-73366  I  ^34 
8-73600  I  ^34 

8-73832 !  ^3^ 

8.74292  1  22Q 
8.74521  I  f^„ 
8.74748  226 
8.74974  22? 
8-75199  g 
8.75423  22! 
8.75645  222 
8.75867  ^^ 
8-76087   ^^° 

«-76306  ^9 

8.76525  217 

8.76742  216 

8.76958  f^\ 

8.77173  2IA 

8-77387  ^^\ 
8.77600  j  \ 

8.7781I  2^1 
8.78022  f^ 
8-78232  f^ 

"^^78649"' S 
8.78855  2^6 

8.79061  ^°° 
8.79266  ^\ 

^•79470  ;g 

8-79673  202 

8.79875  20? 

8.80076  ^^ 

8.80277  ^°^ 
8.80476 
8.80674 


99786 

784 
782 
780 
778 


9.99907 
9.99906 
9.99905 
9.99904 
9-99904 


06554 
584 
613 
642 
671 


99776 

774 
772 
770 
768 


9.99903 
9.99902 
9.99901 
9.99900 
9.99899 


06700 
730 
759 
788 
817 


99766 
764 
762 
760 
758 
756 


9.99898 
9.99898 

9-99897 
9.99896 
9.99895 
9-99894 


06847 
876 
905 
934 
963 
993 


8.80872 
8.81068 
8.81264 
8.81459 
8.81653 
8.81846 
8.82038 
8.82230 
8.82420 
8.82610 
8.82799 
8.82987 

8.83175 
8.83361 

8.83547 
8.83732 
8.83916 
8.84100 
8.84282 
8.84464 


L99 


28060 
27819 
27580 

27341 
27104 


19.081 

18.976 

.871 

.768 

.666 


26868 
26634 
26400 
,26168 
25937 


18.564 
.464 
.366 
.268 
.171 


25708 
25479 
25252 
25026 
24801 


18.075 

17.980 

.886 

•793 
.702 


24577 
24355 
24133 
23913 
23694 


17.611 
.521 
.431 
•343 
.256 


23475 
23258 
23042 
22827 
22613 


17.169 

.084 
16.999 

.915 
.832 


22400 
22189 
21978 
,21768 
21559 


16.750 
.668 
.587 
•507 
.428 


2i35£ 
21145 
20939 

20734 
20530 


16.350 
.272 
•195 
.119 
•043 


20327 
,20125 
,19924 
19723 
19524 


15-969 
•895 
.821 

•748 


19326 
19128 
18932 
18736 
18541 


15-605 
.534 
.464 

•394 
.325 


18347 
18154 
17962 
17770 
17580 


15-257 
.189 
.122 
.056 

14.990 


17390 
17201 
17013 
16825 
16639 


14.924 
.860 
.795 
•732 
.669 


16453 
16268 
16084 
15900 
15718 
15536 


14.606 

.544 
482 
421 
.361 
.301 


Nat.  Cos  Log.    d.    Nat.  Sin  Log.  Nat.CotLog.  c.d.  Log.  Tan  Nat 

86° 


Nat.  Sin  Log.  d.  Nat.  Cos  Log. 


Nat.Tan  Log. 


Log.  Cot  Nat. 


06976 
07005 

034 
063 
092 


8.84358 

8.84539 
8.84718 
8.84897 
8.85075 


07121 
150 
179 
208 
237 


8.85252 
8.85429 
8.85605 
8.85780 
8.85955 


07266 
295 
324 
353 
382 


8.86128 
8.86301 
8.86474 
8.86645 
8.86816 


0741 1 
440 
469 
498 
527 


8.86987 
8.87156 
8.87325 
8.87494 
8.87661 


07556 
585 
614 

643 
672 


8.87829 
8.87995 
8.88161 
8.88326 
8.88490 


07701 
730 
759 
788 
817 


8.88654 
8.88817 
8.88980 
8.89142 
8.89304 


07846 

875 
904 

933 

962 


8.89464 
8.89625 
8.89784 
8.89943 
8.90102 


07991 

08020 

049 

078 

107 


8.90260 
8.90417 
8.90574 
8.90730 
8.90885 


08136 

165 
194 
223 
252 


8.91040 
8.91 195 
8.91349 
8.91502 
8.91655 


08281 
310 
339 
368 
397 


8.91807 
8.91959 
8.921 10 
8.92261 
8.92411 


08426 

455 
484 

513 

542 


8.92561 
8.92710 
8.92859 
8.93007 
8-93154 


08571 

8.93301 

600 

8.93448 

629 

8.93594 

658 

687 

8.93885 

716 

8.94030 

181 
179 
179 

178 
177 
177 

176 

175 
175 
173 
173 
173 
171 
171 
171 

169 
169 
169 
167 

168 
166 
166 

165 

164 
164 

163 

163 

162 
162 
160 

161 
159 
159 
159 
158 

157 
157 

156 
155 
155 
155 
154 
153 
153 
152 
152 
151 
151 
150 
150 
149 
149 
148 
147 
147 
147 

146 
146 

145 
145 


99756 

754 
752 
750 
748 


9.99894 
9.99893 
9.99892 
9.99891 
9.99891 


06993 
07022 

051 

080 


8.84464 
8.84646 
8.84826 
8.85006 
8.85185 


99746 

9.99890 

07139 

8.85363 

744 

9.99889 

8.85540 

742 

9.99888 

197 

8.85717 

740 

9.99887 

227 

738 

9.99886 

256 

8.86069 

99736 

9.99885 

07285 

8.86243 

734 

9.99884 

314 

8.86417 

731 

9.99883 

344 

8.86591 

729 

9.99882 

373 

8.86763 

727 

9.99881 

402 

8.86935 

99725 

9.99880 

07431 

8.87106 

723 

9.99879 

461 

8.87277 

721 

9.99879 

490 

8.87447 

719 

9.99878 

519 

8.87616 

716 

9.99877 

548 

8.87785 

99714 

9.99876 

07578 

8.87953 

712 

9.99875 

607 

8.88120 

710 

9.99874 

636 

8.88287 

70B 

9.99873 

665 

8.88453 

705 

9.99872 

695 

8.88618 

99703 

9.99871 

07724 

8.88783 

701 

9.99870 

753 

8.88948 

699 

9.99869 

782 

8.891 1 1 

696 

9.99868 

812 

8.89274 

694 

9.99867 

841 

8.89437 

99692 

9.99866 

07870 

8.89598 

689 

9.99865 

899 

8.89760 

687 

9.99864 

929 

8.89920 

685 

9.99863 

958 

8.90080 

683 

9.99862 

987 

8.90240 

99680 

9.99861 

08017 

8.90399 

678 

9.99860 

046 

8.90557 

676 

9.99859 

075 

8.90715 

673 

9.99858 

104 

8.90872 

671 

9.99857 

134 

8.91029 

99668 

9.99856 

08163 

8.91 185 

666 

999855 

192 

8.91340 

664 

9.99854 

221 

8.91495 

661 

9.99853 

251 

8.91650 

659 

9.99852 

280 

8.91803 

99657 

9.99851 

08309 

8.91957 

654 

9.99850 

339 

8.92110 

652 

9.99848 

368 

8.92262 

649 

9.99847 

397 

8.92414 

647 

9.99846 

427 

8.92565 

99644 

9.99845 

08456 

8.92716 

642 

9.99844 

485 

8.92866 

639 

9.99843 

514 

8.93016 

637 

9.99842 

544 

8.93165 

635 

9.99841 

573 

8.93313 

99632 
630 
627 

625 

622 
619 


9.99840 
9.99839 
9.99838 
9.99837 
9.99836 
9.99834 


08602 
632 
661 

690 
720 

749 


8.93462 
8.93609 
8.93756 
8.93903 
8.94049 
8.94195 


82 
80 
80 
79 
78 

n 
77 
76 
76 
74 
74 
74 
72 
72 
71 
71 
70 
69 
69 

68 
67 
67 
66 

65 
65 
65 
63 
63 
63 
61 
62 
60 
60 
60 
59 
58 
58 
57 
57 
56 
55 
55 
55 
53 
54 
53 
152 
52 
51 
51 
50 
50 
49 
48 
49 
47 
47 
47 
46 
46 


1.15536 
1.15354 
1.15174 
1.14994 
1.14815 


14.301 
.241 
.182 
.124 
.065 


1.14637 
1.14460 
1.14283 
1.14107 
1.13931 


14.008 

13.951 

.894 

.838 

.782 


1.13757 
1.13583 
1.13409 

1.13237 
1.13065 


13.727 
.672 
.617 

.563 
.510 


1.12894 
1.12723 

1.12553 
1.12384 
1.12215 


13.457 
.404 
.352 
.300 
.248 


1.12047 
1.11880 
1.11713 

1.11547 
1.11382 


13.197 
.146 
.096 
.046 

12.996 


1.11217 
1.11052 
1.10889 
1.10726 
1.10563 


12.947 
.898 
.850 
.801 
.754 


1.10402 
1.10240 
1.10080 
1.09920 
1.09760 


12.706 

.659 
.612 
.566 
.520 


1.09601 
1.09443 
1.09285 
1.09128 
1.08971 


12.474 
.429 
.384 
.339 
.295 


1.08815 
1.08660 
1.08505 
1.08350 
1.08197 


12.251 
.207 
.163 
.120 
.077 


1.08043 
1.07890 
1.07738 
1.07586 
1.07435 


12.035 

11.992 

.950 

.909 


1.07284 
1.07134 
1.06984 
1.06835 
1.06687 


11.826 
.785 
.745 
.705 
.664 


1.06538 
1.06391 
1.06244 
1.06097 

1.05951 
1.05805 


11.625 
.585 
.546 
•507 
.468 
.430 


Nat.  Cos  Log.    d. 


Nat.  Sin  Log.  Nat.  Cot  Log 

86^ 


c.d.  Log.  Tan  Nat. 


'    Nat.  Sin  Log.    d.    Nat.  Cos  Log. 


Nat.Tan  Log. 


c.d. 


Log.  Cot  Nat. 


08716 
745 
774 
803 

831 


8.94030 
8.94174 
8.94317 
8.94461 
8.94603 


08860 
889 
918 

947 
976 


8.94746 
8.94887 
8.95029 
8.95170 
8.95310 


09005 

034 
063 
092 
121 


8.95450 
8.95589 
8.95728 
8.95867 
8.96005 


09150 
179 
208 
237 


8.96143 
8.96280 
8.96417 

8.96553 
8.96689 


09295 
324 
353 
382 
411 


8.96825 
8.96960 
8.97095 
8.97229 
8.97363 


09440 
469 
498 
527 

_556. 

09585 
614 
642 
671 
700 


8.97496 
8.97629 
8.97762 
8.97894 
8.98026 


8.98157 
8.98288 
8.98419 

8.98549 
8.98679 


09729 
758 
707 
816 

845 


09874 
903 
932 
961 
990 


10019 
048 
077 
106 
135 


9.00082 
9.00207 
9.00332 
9.00456 
9.00581 


10164 
192 
221 
250 
279 


9.00704 
9.00828 
9.00951 
9.01074 
9.01196 


10308 

337 
366 

395 
424 

453 


9.01318 
9.01440 
9.01561 
9.01682 
9.01803 
9.01923 


8.98808 

8.98937 
8.99066 
8.99194 
8.99322  I 


8.99450 

8.99577 
8.99704 
8.99830 
8.99956 


99619 
617 
614 
612 
609 


999834 
999833 
9.99832 
9.99831 
9.99830 


08749 
778 
807 

837 
866 


8.94195 
8.94340 
8.94485 
8.94630 
8.94773 


99607 
604 
602 

599 
596 


9.99829 
9.99828 
9.99827 
9.99825 
9.99824 


08895 
925 
954 
983 

09013 


8.94917 
8.95060 
8.95202 
8.95344 
8.95486 


99594 
591 
588 
586 
583 


9.99823 
9.99822 
9.99821 
9.99820 
9.99819 


09042 
071 

lOI 

130 
159 


8.95627 
8.95767 
8.95908 
8.96047 
8.96187 


99580 
578 
575 
572 
570 


9.99817 
9.99816 
9.99815 
9.99814 
9.99813 


09189 
218 
247 
277 
306 


8.96325 
8.96464 
8.96602 

8.96739 
8.96877 


99567 
564 
562 

559 
556 


9.99812 
9.99810 
9.99809 
9.99808 
9.99807 


09335 
365 
394 
423 
453 


8.97013 
8.9715? 
8.97285 
8.97421 
8.97556 


99553 
551 
548 
545 
542 


9.99806 
9.99804 
9.99803 
9.99802 
9.99801 


09482 
5" 
541 
570 
600 


8.97691 
8.97825 

8.97959 
8.98092 
8.98225 


99540 
537 
534 
531 
528 


9.99800 
9.99798 
9.99797 
9.99796 
9.99795 


09629 
658 
688 
717 
746 


8.98358 
8.98490 
8.98622 

8.98753 
8.98884 


99526 
523 
520 
517 
514 


9.99793 
9.99792 
9.99791 
9.99790 
9.99788 


09776 
805 

834 
864 

893 


8.9901$ 
8.9914$ 
8.99275 
8.99405 
8.99534 


995 1 1 
508 
506 
503 
500 


9.99787 
9.99786 

9.99785 
9.99783 
9.99782 


09923 
952 
981 

lOOII 

040 


8.99662 
8.99791 
8.99919 
9.00046 
9.00174 


99497 
494 
491 
488 
485 


9.99781 
9.99780 
9.99778 
9-99777 
9.99776 


10069 
099 

128 
158 
187 


9.00301 
9.00427 
900553 
9.00679 
9.00805 


99482 

479 
476 

473 
470 


9.99775 
9.99773 
9.99772 
9.99771 
9.99769 


102 16 
246 
275 
305 
334 


9.00930 
9.01055 
9.01179 
9.01303 
9.01427 


99467 
464 
461 

458 
455 
452 


9.99768 
9.99767 

9.99765 
9.99764 

999763 
9.99761 


10363 

393 
422 
452 
481 
510 


9.01550 
9.01673 
9.01796 
9.01918 
9.02040 
9.02162 


05805 
05660 
05515 
05370 
05227 


C.430 
.392 
.354 
.316 
.279 


05083 
04940 
04798 
04656 
04514 


.242 
.205 
.168 
.132 
.095 


04373 
04233 
,04092 

03953 
03813 


11.059 

.024 

10.988 

.953 
.918 


03675 
03536 
03398 
,03261 
03123 


10.883 
.848 
.814 
.780 
.746 


,02987 
,02850 
.02715 

■02579 
02444 


10.712 
.678 
.645 

.6X2 

.579 


.02309 
,02175 

,02041 
,01908 

01775 


10.546 
.514 

.481 

•449 
.417 


,01642 
,01510 
,01378 
,01247 
,01116 


10.385 
•354 
.322 
.291 
.260 


.00985 
.00855 
.0072$ 

.00595 
.00466 


10.229 
.199 
.168 
•138 
.108 


.00338 
.00209 
.00081 

0.99954 
0.99826 


10.078 
.048 
.019 

9.9893 
601 


0.99699 
0.99573 
0.99447 
0.99321 
0.99195 


9.9310 
021 

9.8734 
448 
164 


0.99070 
0.98945 
0.98821 
0.98697 
0.98573 


9,7882 
601 
322 
044 

9.6768 


0.98450 
0.98327 
0.98204 
0.98082 
0.97960 
0.97838 


9.6493 
220 

9-5949 
679 
411 
144 


Nat. Cos  Log.    d.    Nat.  Sin  Log.  Nat.  Cot  Log.  c.d.  Log. Tan  Nat.     ' 

84° 


6' 


f 

Nat.  Sin  Log.  d. 

|Nat.CoSLog 

|Nat.Tan  Log. 

[:i 

Log.  Cot  Nat. 

^^ 

0 

10453 

9.01923 

120 

99452  9-99761 

I05IO  9.02162 

0.97838 

9.5144 

60 

I 

482 

9.02043 

120 

449  9-99760 

540  9.02283 

0.97717 

9.4878 

59 

2 

511 

9.02163 

446  9-99759 

569  9.02404 

0.9750 

614 

58 

3 

540 

9.02283 

119 
118 

443  9-99757 

599  9-0252$ 

0.97475 

352 

57 

4 

5^9 

9.02402 

440  9-99756 

628  9.02645 

121 

0.97355 

090 

56 
55 

5 

IOS97 

9.02520 

99437  9-99755 

10657  9.02766 

0.97234 

9-3831 

6 

626 

9.02639 

434  9-99753 

687  9-02885 

0.9711$ 

572 

54 

7 

655 

9.02757 

"^ 

431  9-99752 

716  9.03005 

0.96995 

315 

53 

8 

684 

9.02874 

428  9.99751 

746  9.03124 

118 

0.96876 

060 

52 

9 
10 

713 

9.02992 

117 
117 
116 

424  9-99749 

775  9-03242 

119 
118 

0.96758 

9.2806 

51 
50 

10742 

9-03109 

99421  9.99748 

10805  9-03361 

0.96639 

9.2553 

II 

771 

9.03226 

418  9.99747 

834  9-03479 

118 

0.96521 

302 

49 

12 

800 

9-03342 

415  9-99745 

863  9-03597 

0.96403 

052 

48 

13 

829 

9-03458 

412  9.99744 

893  9-03714 

117 

0.96286 

9.1803 

47 

14 

858 

9-03574 

116 

115 
115 
114 

"5 
"3 
114 
114 
113 

409  9.99742 

922  9.03832 

116 

0.96168 

555 

46 

15 

10887 

9.03690 

99406  9.99741 

10952  9.03948 

0.96052 

9.1309 

45 

lb 

916 

9-03805 

402  9.99740 

981  9.04065 

117 

0.95935 

06s 

44 

17 

945 

9.03920 

399  9-99738 

I  ion  9.04181 

116 

0.95819 

9.0821 

43 

i8 

973 

9.04034 

396  9-99737 

040  9.04297 

116 

0.95703 

579 

42 

19 

1 1002 

9.04149 

393  9.99736 

070  9.04413 

115 

0.95587 

338 

41 
40 

20 

1 103 1 

9.04262 

99390  9.99734 

1 1099  9.04528 

0.95472 

9.0098 

21 

060 

9-04376 

386  9-99733 

128  9.04643 

115 

0.95357 

8.9860 

39 

22 

089 

9-04490 

383  999731 

158  9-04758 

115 

0.95242 

623 

38 

23 

118 

9.04603 

380  9.99730 

187  9-04873 

115 

0.95127 

387 

37 

24 

147 

9.04715 

113 
112 

377  9-99728 

217  9.04987 

114 

114 

0.95013 

152 

36 
35 

25 

11176 

9.04828 

99374  9-99727 

I 1246  9.05101 

0.94899 

8.8919 

2b 

205 

9.04940 

112 

370  9-99726 

276  9.05214 

113 

0.94786 

686 

34 

27 

234 

9.05052 

112 

367  9-99724 

305  9-05328 

0.94672 

455 

33 

28 

2b3 

9.05164 

III 

364  9.99723 

335  9-05441 

113 

0-94559 

225 

32 

29 

30 

291 

9-05275 

III 

360  9-99721 

364  9-05553 

113 

0-94447 

8.7996 

31 
30 

1 1320 

9-05386 

99357  9-99720 

"394  9-05666 

0-94334 

8.7769 

31 

349 

9.05497 

354  999718 

423  9-05778 

0.94222 

542 

29 

32 

378 

9.05607 

351  9-99717 

452  9.05890 

0.94110 

317 

28 

33 

407 

9.05717 

347  9-99716 

482  9.06002 

112 

0.93998 

093 

27 

34 

436 

9-05827 

no 

109 
109 
109 

108 

344  9-99714 

511  9.061 13 

III 

0.93887 

8.6870 

26 
25 

35 

11465 

9-05937 

99341  9-99713 

11541  9.06224 

0.93776 

8.6648 

3t> 

494 

9.06046 

337  9-99711 

570  9-0633$ 

427 

24 

37 

523 

9.06155 

334  9-99710 

600  9.06445 

0.93555 

208 

23 

3a 

552 

9.06264 

331  9-99708 

629  9.06556 

0.93444 

8.5989 

22 

39 
40 

580 

9-06372 

109 

327  9-99707 

659  9.06666 

109 

0.93334 

772 

21 
20 

1 1609 

9.06481 

99324  9-99705 

I 1688  9.06775 

0.9322$ 

8.5555 

41 

638 

9.06589 

107 
108 

320  9-99704 

718  9.06885 

no 

0.931 15 

340 

iq 

42 

667 

317  9.99702 

747  9-06994 

109 
109 
108 
109 
108 
108 

0.93006 

126 

18 

43 

696 

9.00004 

107 
107 
106 

314  9.99701 

777    907103 

0.92897 

8.4913 

17 

44 

725 

9.0691 1 

310  9-99699 

806  9.07211 

0.92789 

701 

16 
15 

45 

1 1754 

9.07018 

99307  9.99698 

1 1 836  9.07320 

0.92680 

8.4490 

4b 

783 

9.07124 

303  9.99696 

865  9,07428 

0.92572 

280 

14 

47 

812 

9.07231 

107 

300  9.99695 

895  9-07536 

0.92464 

071 

13 

48 

840 

9-07337 

105 
106 

105 

297  999693 

924  9.07643 

107 

tdR 

0.92357 

8.3863 

12 

49 

8b9 

9.07442 

293  9.99692 

954  9-07751 

107 
106 

0.92249 

656 

II 

50 

11898 

9.07548 

99290  9.99690 

1 1983  9.07858 

0.92142 

8.3450 

10 

51 

927 

9-07653 

286  9.99689 

12013  9.07964 

0.92036 

24s 

9 

52 

956 

9-07758 
9-07863 

105 

283  9.99687 

042  9.08071 

107 
106 
T06 
106 
106 

0.91929 

041 

8 

53 

985 

105 

279  9.99686 

072  9.08177 

0.91823 

8.2838 

7 

54 

12014 

9.07968 

105 
104 

276  9.99684 

loi  9.08283 

0.91717 

636 

6 

55 

12043 

9.08072 

99272  9.99683 

12131  9.08389 

0.91611 

8.2434 

5 

5t> 

071 

9.08176 

^269  9.99681 

160  9.08495 

0.91505 

234 

4 

57 

100 

9.08280 

265  9.99680 

190  9.08600 

105 

0.91400 

035 

3 

5a 

129 

9-08383 

103 

262  9.99678 

219  9.08705 

105 

0.91295 

8.1837 

2 

|g 

158 

9.08486 

103 

258  9.99677 

249  9.08810 

105 

0.91190 

640 

I 

187 

9-08589 

103 

255  9-99675 

278  9.08914 

104 

0.91086 

443 

0 

Nat.  Cos  Log.  d. 

Nat.  Sin  Log. 

Nat.  Cot  Log. 

C.d. 

Log.  Tan  Nat. 

f 

83^ 


f 

Nat.  S 

in  Log. 

d. 

Nat.  Cos  Log. 

Nat.Tan  Log. 

c.d. 

Log.  Cot  Nat. 

" 

0 

12187 

9.08589 

103 
103 
102 

99255 

999675 

12278 

9.08914 

105 

0.91086 

8.1443 

60 

I 

216 

9.08692 

251 

9-99674 

308 

9.09019 

0.90981 

248 

5Q 

2 

245 

9.08795 

248 

9-99672 

.33B 

9.09123 

104 

Tr\A 

0.90877 

054 

58 

3 

274 

9.08897 

244 

9.99670 

3(V 

9.09227 

103 
104 
103 

0.90773 

8.0860 

57 

4 

302 

9.08999 

102 

lOI 

240 

9.99669 

397 

9.09330 

0.90670 

667 

56 

1233 1 

9.09101 

99237 

9.99667 

12426 

9.09434 

0.90566 

8.0476 

55 

6 

360 

9.09202 

102 

233 

9.99666 

456 

9.09537 

0.90463 

28:; 

54 

7 

3»9 

9.09304 

lOI 

230 

9.99664 

485 

9.09640 

0.90360 

095 

53 

8 

418 

9.09405 

226 

9.99663 

515 

9.09742 

103 
102 

0.90258 

7.9906 

52 

9 

447 

9.09506 

100 

222 

9.99661 

544 

9.09845 

0.90155 

718 

51 
50 

10 

12476 

9.09606 

99219 

9.99659 

12574 

9.09947 

0.90053 

7.9530 

II 

504 

9.09707 

100 

215 

9.99658 

603 

9.10049 

0.89951 

344 

49 

12 

9.09807 

100 

211 

9.99656 

633 

9.10150 

0.89850 

158 

48 

13 

562 

9.09907 

99 
100 

99 
99 
98 

99 
98 
98 
98 
98 
97 
97 
97 

208 

9-99655 

662 

9.10252 

0.89748 

7.8973 

47 

14 
15 

591 

9.10006 

204 

9.99653 

692 

9-10353 

lOI 
lOI 

0.89647 

789 

46 

12620 

9.10106 

99200 

9.99651 

12722 

9.10454 

0.89546 

7.8606 

45 

lb 

649 

9.10205 

197 

9.99650 

751 

0.89445 

424 

44 

17 

678 

910304 

193 

9.99648 

781 

9.10656 

0.89344 

243 

43 

18 

70b 

9.10402 

189 

9.99647 

810 

9.10756 

0.89244 

062 

42 

19 

735 

9.10501 

186 

9.99645 

840 

9.10856 

100 

100 

0.89144 

7.7882 

41 

20 

12764 

9.10599 

99182 

9.99643 

12869 

9.10956 

0.89044 

7.7704 

40 

21 

793 

9.10697 

178 

9-99642 

899 

9. 1 1056 

99 
99 
99 
99 

^2 

^0 
98 

98 

98 

97 

0.88944 

525 

39 

22 

822 

9.10795 

175 

9.99640 

929 

9.11155 

0.88845 

.348 

38 

23 

8S1 

9-10893 

171 

9.99638 

958 

9.11254 

0.88746 

171 

37 

24 

880 

9.10990 

ib7 

999637 

988 

9.11353 

0.88647 

7.6996 

36 

25 

12908 

9. 1 1087 

99163 

9-99635 

13017 

9.11452 

0.88548 

7.6821 

35 

2b 

937 

9.1 1 184 

160 

9-99633 

047 

9.11551 

0.88449 

647 

34 

27 

966 

9.11281 

97 
96 

97 
96 
96 

li 
95 
95 
95 

i5f> 

9.99632 

076 

9.11649 

0.88351 

473 

33 

28 

995 

9- "377 

152 

9-99630 

106 

9.11747 
9.11845 

0.88253 

301 

32 

29 

13024 

9.11474 

148 

9.99629 

r3t 

0.88155 

129 

31 

30 

13053 

9.1 1570 

99144 

9-99627 

13165 

9.11943 

0.88057 

7.5958 

30 

31 

081 

9.11666 

141 

9.99625 

195 

9.12040 

0.87960 

787 

29 

32 

no 

9.11761 

137 

9.99624 

224 

9.12138 

0.87862 

618 

28 

33 

139 

9.11857 

133 

9.99622 

254 

9.12235 

0.87765 

449 

27 

34 

168 

9.11952 

129 

9.99620 

284 

9.12332 

96 

96 
96 

95 
95 
95 
95 
95 
94 
95 
94 
94 
93 
94 
93 
93 
93 
93 
92 
92 

93 
91 
92 

0.87668 

281 

26 

35 

13197 

9.12047 

99125 

9.99618 

13313 

9.12428 

0.87572 

7.5113 

25 

3^ 

226 

9.12142 

122 

9.99617 

343 

0.87475 

7.4947 

24 

37 

254 

9.12236 

95 
94 
94 
93 

118 

9-99615 

372 

9.12621 

0.87379 

781 

23 

3a 

283 

9.12331 

114 

9.99613 

402 

9.12717 

0.87283 

(515 

22 

39 

312 

9.12425 

no 

9.99612 

432 

9.12813 

0.87187 

451 

21 
20 

40 

13341 

9.12519 

99106 

9.99610 

13461 

9.12909 

0.87091 

7.4287 

41 

370 

9.12612 

102 

9.99608 

491 

9.13004 

0.86996 

124 

19 

42 

399 

9.12706 

098 

9.99607 

521 

9.13099 

0.86901 

7.3962 

18 

43 

427 

9.12799 

93 
93 
93 
93 

094 

9.99605 

550 

9.13194 

0.86806 

800 

17 

44 

456 

9.12892 

091 

9.99603 

580 

9.13289 

0.86711 

639 

lb 

45 

13485 

912985 

99087 

9.99601 

13609 

9.13384 

0.86616 

7.3479 

15 

46 

514 

9.13078 

083 

9.99600 

639 

9.13478 

0.86522 

319 

14 

47 

543 

9.13171 

93 
92 
92 
92 
92 

079 

9.99598 

bb9 

9.13573 

0.86427 

160 

13 

48 

572 

9.13263 

075 

9.99596 

b98 

9.13667 

0.86333 

002 

12 

49 

600 

9-13355 

071 

9-99595 

728 

9.13761 

0.86239 

7.2844 

II 

50 

13629 

9.13447 

99067 

9.99593 

13758 

9.1385+ 

0.86146 

7.2687 

10 

^i 

658 

063 

787 

9.13948 

0.86052 

531 

9 

52 

687 

9.13630 

91 
92 

059 

9.99589 

817 

9.14041 

0.85959 

375 

8 

ss 

716 

9.13722 

055 

9.99588 

846 

9.14134 

0.85866 

220 

7 

54 

744 

913813 

91 
91 

051 

9.99586 

876 

9.14227 

0.85773 

066 

6 

55 

13773 

9.13904 

99047 

9-99584 

13906 

9.14320 

0.85680 

7.1912 

5 

.0 

802 

9-13994 

043 

9.99582 

935 

9.14412 

0.85588 

759 

4 

57 

831 

9-14085 

91 

039 

9.99581 

965 

9.14504 

0.85496 

607 

3 

5a 

860 

9-14175 

90 
91 

035 

9-99579 

995 

9.14597 

0.85403 

455 

2 

^0 

889 

9.14266 

031 

999577 

14024 

9.14688 

0.85312 

304 

1 

917 

9-14356 

90 

027 

9-99575 

054 

9.14780 

0.85220 

X54 

0 

Nat.  Cos  Log. 

d. 

Nat.  Sin  Log. 

Nat.  Cot  Log. 

c.d. 'Log. Tan  Nat. 

r 

82^ 


8= 


r 

Nat.  Sin  Log.  d. 

Nat.  Cos  Log 

Nat.Tan  Log. 

c.d. 

Log.  Cot  Nat. 

0 

13917  9-I4356 

89 
90 
89 
90 
89 
88 

99027  9.99575 

14054 

9.14780 

0.85220  7.1154 

60 

I 

946  9-14445 

023  9-99574 

084 

91 
91 
91 
91 
91 

0.85128   004 

59 

2 

975  9.14535 

019  9-99572 

113 

9.14963 

0.85037  7.0855 

58 

3 

14004  9.14624 

015  9-99570 

143 

9.15054 

57 

4 
5 

033  9.14714 

on  9.99568 

173 

9.15145 

0.84855   558 

56 
55 

1406 I  9.14803 

99006  9.99566 

14202 

9.15236 

0.84764  7.0410 

b 

090  9.14891 

89 
89 
88 

002  9.99565 

232 

9.15327 

0.84673   264 

54 

7 

119  9.14980 

98998  9.99563 

262 

9.15417 

90 

0.84583    117 

53 

8 

148  9.15069 

994  9-99561 

291 

9.15508 

91 
90 
90 
89 

90 
89 
89 
88 

0.84492  6.9972 

52 

9 

177  9-15157 

88 
88 

990  9.99559 

321 

9.15598 

0.84402   827 

51 

10 

14205  9-15245 

98986  9.99557 

14351 

9.15688 

0.84312  6.9682 

50 

II 

234  9-15333 

88 

982  9.99556 

381 

9.15777 

0.84223   538 

49 

12 

263  9.15421 

87 
88 

978  9-99554 

410 

9.15867 

0.84133   395 

48 

13 

292  9-15508 

973  9-99552 

440 

9.15956 

0.84044   252 

47 

14 

320  9.15596 

87 
87 

f7 
87 
86 

969  9.99550 

470 

9.16046 

0.83954    110 

46 

15 

14349  9-15683 

98965  9.99548 

14499 

9.16135 

0.83865  6.8969 

45 

lb 

378  9-15770 

961  9.99546 

529 

9.16224 

0.83776   828 

44 

17 

407  9-15857 

957  9-99545 

559 

9.16312 

89 
00 

0.83688   687 

43 

i8 

436  9-15944 

953  9-99543 

588 

9.16401 

0.83599   548 

42 

19 

464  9.16030 

86 
87 
86 

948  9-99541 

618 

9.16489 

88 
88 

0.83511    408 

41 

20 

14493  9.16116 

98944  9.99539 

14648 

9.16577 

0.83423  6.8269 

40 

21 

522  9.16203 

940  9-99537 

678 

88 

0.83335    131 

■?9 

22 

551  9-16289 

'4 

936  9-99535 

707 

9.16753 

88 

0.83247  6.7994 

38 

23 

580  9.16374 

931  9.99533 

737 

9.16841 

87 
88 

87 
87 
87 

Of. 

0.83159   856 

37 

24 

608  9.16460 

85 

86 

927  9-99532 

767 

9.16928 

0.83072   720 

.36 

25 

14637  9.16545 

98923  9-99530 

14796 

9.17016 

0.82984  6,7584 

35 

2b 

666  9.16631 

85 
85 
84 

85 
84 
84 
84 
84 

?3 
84 

^3 
83 

83 
83 
83 
82 

919  9-99528 

826 

9.17103 

0.82897   448 

34 

27 

695  9.16716 

914  9-99526 

856 

9.17190 

0.82810   313 

33 

28 

723  9.16801 

910  9.99524 

886 

9.17277 

0.82723    179 

32 

29 

752  9.16886 

906  9.99522 

915 

9.17363 

87 

86 

0.82637   045 

31 

1478 I  9.16970 

98902  9.99520 

1494s 

9.17450 

0.82550  6.6912 

-3-0 

31 

810  9.17055 

897  9.99518 

975 

9-17536 

86 

0.82464   779 

29 

32 

838  9-17139 

893  9.99517 

15005 

9.17622 

86 

0.82378   646 

28 

33 

867  9.17223 

889  9.99515 

034 

9.17708 

86 

0.82292   514 

27 

34 

896  9.17307 

884  9.99513 

064 

9.17794 

86 
85 

0.82206   383 

26 
25 

35 

14925  9.17391 

98880  9.99511 

15094 

9.17880 

0.82120  6.6252 

3^ 

954  9-17474 

876  9.99509 

124 

9-17965 

0.82035    122 

24 

37 

982  9.17558 

871  9.99507 

153 

9.18051 

85 
85 
85 

84 

l^ 
84 

84 

84 

84 

83 

84 

83 

83 
83 
Ro 

0.81949  6.5992 

23 

38 

15011  9.17641 

867  9.99505 

183 

9.18136 

0.81864   863 

22 

39 

040  9.17724 

863  9.99503 

213 

9.18221 

0-81779    734 

21 
20 

40 

15069  9.17807 

98858  9.99501 

15243 

9.18306 

0.81694  6.5606 

41 

097  9.17890 

854  9.99499 

272 

9.18391 

0.81609    478 

19 

42 

126  9.17973 

849  9.99497 

302 

0.81525    350 

18 

43 

155  9-18055 

82 

845  9.99495 

332 

9.18560 

0.81440    223 

17 

44 

184  9.18137 

83 
82 

841  9.99494 

362 

9.18644 

0.81356    097 

16 

45 

15212  9.18220 

98836  9.99492 

15391 

9.18728 

0.81272  6.4971 

15 

46 

241  9.18302 

81 

832  9.99490 

421 

9.18812 

0.81188    846 

14 

47 

270  9.18383 

82 

827  9.99488 

451 

9.18896 

0.81104    721 

13 

48 

299  9.18465 

82 

823  9.99486 

481 

9.18979 

0.81021    596 

12 

49 

327  9.18547 

81 
81 

818  9.99484 

511 

9.19063 

0.80937    472 

11 

50 

15356  9.18628 

98814  9.99482 

15540 

9.19146 

0.80854  6.4348 

10 

SI 

385  9.18709 

81 

809  9.99480 

570 

9.19229 

0.80771    225 

9 

52 

414  9-18790 

81 

805  9.99478 

600 

9.19312 

0.80688    103 

8 

S3 

442  9.18871 

81 

800  9.99476 

630 

9.19395 

0.80605  6.3980 

7 

54, 
55 

471  9-18952 

81 
80 

796  9.99474 

660 

9.19478 

0.80522    859 

6 

15500  9.19033 

98791  9-99472 

15689 

9.19561 

0.80439  6.3737 

5 

Sb 

529  9.19113 

80 

787  9-99470 

719 

9.19643 

82 

0.80357    617 

4 

S7 

557  919193 

80 

782  9.99468 

749 

9-19725 

82 
82 
82 

0.80275    496 

3 

S8 

586  9-19273 

80 

778  9.99466 

779 

9.19807 

0.80193    376 

2 

IS 

615  9-19353 

80 

773  9.99464 

809 

9.19889 

0.80111    257 

1 

643  9-19433 

769  9.99463 

838 

9.19971 

0.80029    138 

0 

Nat.  Cos  Log.  d. 

Nat.  Sin  Log. 

Nat.  Cot  Log. 

C.d.  Log.  Tan  Nat. 

f 

81° 


r 

Nat.  Sin  Log.  d.  1 

Nat.  Cos  Log. 

Nat.Tan  Log.| 

c.d. 

Log.  Cot  Nat. 

0 

15643  9.19433 

80 

98769 

9.99462 

15838 

9.19971 

82 

0.80029  6.3138 

60 

I 

672  9.I95I3 

79 

764 

9.99460 

868 

9-20053 

Rt 

0-79947   019 

59 

2 

701  9-19592 

760 

9-99458 

898 

9-20134 

80 

0.79866  6.2901 

58 

3 

730  9.19672 

79 
79 
79 

755 

9-99456 

928 

9.20216 

81 

0.79784   783 

57 

4 
5 

758  9.I975I 
15787  9.19830 

751 

9-99454 

958 

9.20297 

8i 
8t 

0.79703    666 

56 
55 

98746 

999452 

15988 

9-20378 

0.79622  6.2549 

b 

816  9.19909 

741 

9-99450 

I60I7 

9.20459 

0.79541    432 

54 

7 

845  9.19988 

79 

737 

9.99448 

047 

9.20540 

81 
80 

0.79460    316 

S3 

8 

873  9.20067 

78 
78 

79 
78 
78 
77 
78 
78 
77 

732 

9.99446 

077 

9.20621 

0-79379   200 

52 

9 
10 

902  9.20145 

728 
98723 

9-99444 
9.99442 

107 

9.20701 

81 
80 

0.79299   085 

51 
50 

1593 1  9.20223 

I6I37 

9.20782 

0.79218  6.1970 

II 

959  9.20302 

718 

9.99440 

167 

9.20862 

80 
80 
8n 

0.79138    856 

49 

12 

988  9.20380 

714 

999438 

196 

9.20942 

0.79058   742 

48 

13 

16017  9.20458 

709 

9-99436 

226 

9.21022 

0.78978   628 

47 

14 

046  9.20535 

704 

9-99434 

25b 

9.21102 

80 

0.78898   515 

46 

15 

16074  9-20613 

98700 

999432 

16286 

9.21182 

0.78818  6.1402 

45 

16 

103  9.20691 

695 

9.99429 

316 

9.21261 

79 
80 

0.78739   290 

44 

17 

132  9.20768 

690 

9.99427 

346 

9-21341 

0.78659    178 
0.78580   066 

43 

lb 

160  9.20845 

77 
77 
77 

686 

999425 

376 

9.21420 

79 
79 
79 

42 

19 
20 

189  9.20922 

681 

999423 

405 

9.21499 

0.78501  6.0955 

41 
40 

16218  9.20999 

98676 

9.99421 

16435 

9.21578 

0.78422  6.0844 

21 

246  9.21076 

671 

9.99419 

465 

9.21657 

79 

0.78343    734 

39 

22 

275  9.21153 

\l 

667 

9.99417 

495 

9.21736 

79 
78 

0.78264    624 

38 

23 

304  9.21229 

662 

9-99415 

525 

9.21814 

0.78186    514 

37 

24 

333  9.21306 

77 
76 

7^ 

76 

76 

657 
98652 

9-99413 

555 

9.21893 

79 

78 

78 
78 
78 
78 
78 

0.78107    405 

36 
35 

25 

16361.  9.21382 

9-994" 

16585 

9.21971 

0.78029  6.0296 

2b 

390  9.21458 

648 

9.99409 

615 

9.22049 

0.77951    188 

34 

27 

419  921534 

643 

9-99407 

645 

9.22127 

0.77873    080 

33 

28 

447  9.21610 

638 

9.99404 

674 

9.22205 

0.77795  5.9972 

32 

29 

476  9.21685 

75 
76 

__633 
98629 

9.99402 

704 

9.22283 

0.77717    865 

31 
30 

30 

16505  9.21761 

9-99400 

16734 

9.22361 

0.77639  5.9758 

31 

533  9.21836 

75 
76 

624 

9.99398 

764 

9.22438. 

77 

78 

0.77562    651 

29 

32 

562  9.21912 

619 

9.99396 

794 

9.22516 

0.77484    545 

28 

33 

591  9.21987 

614 

9-99394 

824 

9-22593 

0.77407    439 

27 

34 

620  9.22062 

75 
75 

609 

9-99392 

854 

9.22670 

77 
77 

0.77330    333 

26 

35 

16648  9.22137 

98604 

9-99390 

16884 

9.22747 

0.77253  5-9228 

25 

3^ 

677  9.22211 

74 

600 

9.99388 

914 

9.22824 

77 

0.77176    124 

24 

37 

706  9.22286 

75 
75 

595 

9-99385 

944 

9.22901 

77 
76 

0.77099    019 

23 

3» 

734  9-22361 

590 

999383 

974 

9.22977 

0.77023  5-8915 

22 

39 

763  9.22435 

74 
74 

585 

9.99381 

17004 

9-23054 

76 
76 

0.76946    811 

21 
20 

40 

16792  9.22509 

98580 

9-99379 

17033 

9.23130 

0.76870  5.8708 

41 

820  9.22583 

74 

575 

063 

9.23206 

o.7%94   605 

19 

42 

849  9-22657 

74 

570 

9.99375 

093 

9.23283 

77 
76 
76 

75 
76 

0.76717   502 

18 

43 

878  9.22731 

74 

565 

9-99372 

123 

9-23359 

0.76641    400 

17 

44 
45 

906  9.22805 
16935  9-22878 

74 
73 

561 

9-99370 

153 

923435 

0.76565   298 

15 

98556 

9.99368 

17183 

9.23510 

0.76490  5.8197 

15 

46 

964  9.22952 

74 

551 

9-99366 

213 

0.76414   095 

14 

47 

992  9.23025 

73 

546 

9-99364 

243 

9.23661 

75 
76 

75 
75 
75 
75 
75 
74 
75 
74 
75 
74 
74 

0.76339  5-7994 

13 

4« 

17021  9.23098 

73 

541 

9.99362 

273 

9-23737 

0.76263    894 

12 

49_ 
50 

050  9.23171 

73 
73 

536 

9.99359 

303 

9.23812 

0.76188    794 

II 

17078  9.23244 

98531 

9.99357 

17333 

9.23887 

0.76113  5.7694 

10 

51 

107  9-23317 

73 
73 
72 

526 

9.99355 

363 

9.23962 

0.76038    594 

9 

52 

136  9-23390 

521 

9-99353 

393 

9.24037 

0.75963    495 
0.75888    396 

8 

S3 

164  9.23462 

516 

9-99351 

423 

9.24112 

7 

54 

193  9-23.'535 

73 
72 

■511 

9.99348 

453 

9.24186 

0.75814    297 

6 

55 

17222  9.23607 

98506 

9-99346 

17483 

9.24261 

0.75739  5.7199 

5 

56 

250  9.23679 

72 
73 

.501 

9-99344 

513 

9-24335 

0.75665    lOI 

4 

57 

279  9-23752 

496 

9-99342 

543 

9.24410 

0.75590    004 

3 

58 

308  9.23823 

71 

491 

9-99340 

573 

9.24484 

0.75516  5.6906 

2 

^0 

336  9.23895 

72 

486 

9-99337 

603 

9-24558 

0.75442    809   11 

365  9-23967 

72 

481 

9-99335 

633 

9.24632  1  '-* 

0.75368   713  o| 

Nat.  Cos  Log.  d. 

Nat.  Sin  Log. 

Nat.  Cot  Log. 

c.d.!  Log.  Tan  Nat. 

LJ 

80' 


10 

0 

r 

Nat.  Sin  Log.  d. 

Nat.  Cos  Log.  d. 

Nat.TanLog. 

c.d 

Log.  Cot  Nat. 

r 

0 

17365  9-23967 

72 
71 
71 
72 
71 
71 
71 
70 

71 
70 

71 
70 
70 
70 
70 
70 

69 
70 
69 

i^ 
69 
69 
69 
69 

6S 

98481  9-99335 

17633  9.24632 

0-75368  5-6713 

60 

I 

393  9-24039 

476  9-99333 

2 

663  9.24706 

74 
73 

0.75294   617 

59 

2 

422  9.241 10 

471  9-99331 

693  9.24779 

0.75221   521 

■;8 

3 

451  9-24181 

466  9.99328 

723  9.24853 

74 

0.75147   425 

^7 

4 

479  9-24253 

461  9.99326 

2 

753  9.24926 

71 
74 

0.75074   329 
0.75000  5.6234 

56 
55 

5 

17508  9-24324 

98455  9.99324 

17783  9.25000 

6 

537  9.24395 

450  9.99322 

3 
2 

813  9.25073 

IZ 

0.74927   140 

S4 

7 

565  9.24466 

445  9.99319 

843  9.25146 

73 

0.74854   045 

53 

8 

594  9-24536 

440  9.99317 

873  9.25219 

0.74781  5.5951 

S2 

9 

623  9.24607 

435  9-99315 
98430  9.99313 

2 
3 

903  9.25292 

73 

0.74708   857 
0.74635  5.5764 

51! 

10 

17651  9.24677 

17933  9.25365 

II 

680  9.24748 

425  9-99310 

963  9.25437 

72 

0.74563   671 

49  i 

12 

708  9.24818 

420  9-99308 

^ 

993  9.25510 

Tb 

0.74490   578 

48' 

i3 

737  9-24888 

414  999306 

" 

18023  9-25582 

0.74418   485 

47 

H 

766  9.24958 

409  9.99304 

3 
2 

053  9-25655 

73 
72 
72 

0.74345   393 

46 

15 

17794  9-25028 

98404  9.99301 

18083  9-25727 

0.74273  5.5301 

45 

lb 

823  9.25098 

399  9.99299 

"3  9-25799 

0.74201   209 

44 

17 

852  9.25168 

394  9.99297 

3 

143  9-25871 

72 

0.74129   118 

43 

l8 

880  9.25237 

389  9.99294 

173  9.25943 

72 

0.74057   026 

42 

19 

909  9-25307 

383  9.99292 

2 

203  9.26015 

72 
71 

0.73985  5-4936 

41 

20 

17937  9-25376 

98378  9.99290 

18233  9.26086 

0.73914  5-4845 

40 

21 

966  9.25445 

373  9.99288 

3 

263  9.26158 

7^ 

0.73842   755 

39 

22 

995  9.25514 

368  9.99285 

293  9.26229 

0.73771   665 

38 

23 

18023  9.25583 

362  9.99283 

"^ 

323  9.26301 

0.73699   575 

37 

24 
25 

052  9.25652 

357  9.99281 

3 

353  9.26372 

71 

0.73628   486 

36 

1808 I  9.25721 

98352  9.99278 

18384  9.26443 

0.73557  5-4397 

35 

2b 

109  9.25790 

347  9-99276 

2 

414  9.26514 

71 

0.73486   308 

34 

27 

138  9-25858 

69 
68 

341  9.99274 

3 
2 

444  9.26585 

0.73415   219 

33 

28 

166  9.25927 

336  9.99271 

474  9.26655 

70 
71 
71 

0.73345   131 

32 

29 

195  9-25995 

68 
68 

331  9.99269 

2 
3 

504  9.26726 

0.73274   043 

31 

30 

18224  9.26063 

98325  9.99267 

18534  9-26797 

0.73203  5-3955 

30 

31 

252  9.26131 

68 

320  9.99264 

564  9.26867 

0.73133   868 

29 

32 

281  9.26199 

68 

315  9.99262 

" 

594  9.26937 

71 

0.73063   781 

28 

33 

309  9.26267 

68 

310  9.99260 

3 
2 

3 
2 

624  9.27008 

0.72992   694 

27 

34 

338  9.26335 

68 
67 
68 

304  9-99257 

654  9.27078 

70 

0.72922   607 

26 

35 

18367  9.26403 

98299  9.99255 

18684  9.27148 

0.72852  5.3521 

25 

3^ 

395  9-26470 

294  9.99252 

714  9.27218 

70 
69 

0.72782   435 

24 

37 

424  9.26538 

67 
67 
67 
67 
67 
67 
67 
66 
67 
66 

288  9.99250 

745  9.27288 

0.72712   349 

23 

3» 

452  9.26605 

283  9.99248 

3 
2 

775  9-27357 

0.72643   263 

22 

39 
40 

481  9.26672 

277  9.99245 
98272  9.99243 

805  9.27427 

70 
69 

69 
69 
69 
69 
69 

68 

0-72573   178 

21 
20 

18509  9-26739 

18835  9.27496 

0.72504  5.3093 

41 

538  9.26806 

267  9.99241 

865  9.27566 

0.72434   008 

19 

42 

567  9.26873 

261  9.99238 

3 

895  9.27635 

0-72365  5-2924 

18 

43 

595  9-26940 

256  9.99236 

3 
2 

925  9-27704 

0.7220   839 

17 

44 
45" 

624  9.27007 

250  9-99233 

955  9.27773 

0.72227   755 

lb 

18652  9.27073 

98245  9.99231 

18986  9.27842 

0.72158  5.2672 

15 

4b 

681  9.27140 

240  9.99229 

3 

19016  9.2791 1 

0.72089   588 

14 

47 

710  9.27206 

67 
66 

234  9.99226 

046  9.27980 

0.72020   505 

13 

48 

738  9.27273 

229  9.99224 

3 
2 

076  9.28049 

0.71951   422 

12 

49 

7(y7  9-27339 

66 
66 

223  999221 

106  9.28117 

69 

68 

0.71883   339 

n 

50 

18795  9.2740g 

98218  9.99219 

19 136  9.28186 

0.71814  5.2257 

10 

51 

824  9.27471 

66 

212  9.99217 

3 

166  9.28254 

0.71746   174 

9 

52 

852  9.27537 

% 

207  9.99214 

197  9.28323 

0.71677   092 

8 

53 

881  9.27602 

20I  9.99212 

3 
2 

3 

227  9.28391 

68 

0.71609   on 

7 

54 

910  9.27668 

66 

65 
65 
66 

196  9.99209 

257  9.28459 

68 
6H 

0.71541  5.1929 

6 

55 

18938  9-27734 

98190  9.99207 

19287  9.28527 

9.71473  5.1848 

5 

5^^ 

967  9.27799 

185  9.99204 

317  9-28595 

67 

68 

0.71405   767 

4 

57 

995  9-27864 

179  9.99202 

347  9.28662 

0.71338   686 

3 

5« 

19024  9.27930 

65 
65 

174  9.99200 

3 

378  9.28730 

68 

0.71270   606 

2 

il 

052  9.27995 

168  9-99197 

408  9.28798 

67 

0.71202   526 

I 

081  9.28060 

163  9-99195 

438  9.28865 

0.7113S   446 

0 

Nat.  Cos  Log.  d.  1 

Nat.  Sin  Log.  d.  1 

Nat.  Cot  Log. 

c.d. 

Log.TanNat. 

/ 

79' 


ir 


r 

Nat.  Sin  Log.  d. 

Nat.  Cos  Log 

d. 

Nat.TanLog. 

c.d. 

Log.  Cot  Nat. 

0 

1908 1  9.28060 

1 

64 
65 
65 
64 

64 

65 
64 

64 

64 

64 

63 
64 
64 
63 
63 
64 
63 
63 
63 

98163  9.99195 

3 
2 

19438  9.28865 

68 

0.71 135  5.1446 

60 

I 

109  9.28125 

157  9-99192 

468  9.28933 

67 
67 
67 
67 
67 
6(S 

0.71067   366 

5Q 

2 

138  9.28190 

152  9.99190 

3 

498  9.29000 

0.71000   286 

58 

3 

167  9.28254 

146  9-99187 

529  9.29067 

0.70933   207 

57 

4 

195  9.28319 

140  9-99185 

3 

559  9-29134 

0.70866   128 

56 

5 

19224  9.28384 

98135  9-99182 

19589  9.29201 

0.70799  5.1049 

55 

6 

252  9.28448 

129  9.99180 

3 

619  9.29268 

0.70732  5.0970 

54 

7 

281  9.28512 

124  9.99177 

649  9.29335 

0.70665   892 

53 

8 

309  9.28577 

118  9-99175 

3 
2 

3 

680  9.29402 

0.70598   814 

S2 

9 

338  9.28641 

112  9-99172 

710  9.29468 

67 
66 

0.70532   736 

51 
50 

10 

19366  9.28705 

98107  9-99170 

19740  9.29535 

0.70465  5.0658 

II 

395  9.28769 

loi  9.99167 

770  9.29601 

67 

66 

0.70399   581 

4Q 

12 

423  9.28833 

096  9.99165 

3 

801  9.29668 

0.70332   504 

48 

13 

452  9.28896 

090  9.99162 

831  9-29734 

66 

0.70266   427 

47 

14 
15 

481  9.28960 

084  9.99160 
98079  9.99157 

3 

861  9.29800 

66 
66 
66 

0.70200   350 

46 
45 

19509  9.29024 

19891  9.29866 

0.70134  5.0273 

I6 

538  9.29087 

^3   9-99155 

3 

921  9.29932 

0.70068   197 

44 

17 

566  9.29150 

067  9.99152 

952  9.29998 

66 

0.70002   121 

43 

l8 

595  9-29214 

061  9.99150 

3 
2 

3 

982  9.30064 

66 

0.69936   045 

42 

19 

623  9.29277 

056  9-99147 

20012  9.30130 

65 
66 

0.69870  4.9969 

41 

20 

19652  9.29340 

98050  9.99145 

20042  9.30195 

0.69805  4.9894 

40 

21 

680  9.29403 

044  9.99142 

073  9.30261 

^5 

0.69739   819 

39 

22 

709  9.29466 

039  9.99140 

3 

103  930326 

0.69674   744 

38 

23 

737  9-29529 

033  9-99137 

133  9-30391 

0.69609   669 

37 

24 

25 

766  9-29591 

63 
60 

027  9-99135 

3 

164  9-30457 

65 

65 
64 

% 
% 

64 

i^ 
64 

64 

64 

64 

i^ 
64 

64 

63 
64 
63 
63 
63 
63 

^3 
63 

63 
62 

0.69543   594 

3b 

19794  9-29654 

98021  9.99132 

20194  9-30522 

0.69478  4.9520 

35 

2b 

823  9.29716 

53 

016  9.99130 

3 
3 

224  9.30587 

0.69413   446 

34 

27 

851  9.29779 

010  9.99127 

254  9-30652 

0.69348   372 

33 

28 

880  9.29841 

6^ 

004  9.99124 

285  9.30717 

0.69283   298 

32 

29 
30 

908  9-29903 

63 
60 

97998  9.99122 

3 

315  9.30782 

0.69218   225 

31 

19937  9-2996<5 

97992  9-99119 

20345  9.30846 

0.69154  4.9152 

30 

31 

965  9.30028 

6'> 

987  9.99117 

3 
2 

376  9.30911 

0.69089   078 

29 

32 

994  9-30090 

61 

981  9.99114 

406  9.30975 

0.69025   006 

28 

33 

20022  9.30151 

6'^ 

975  9-99112 

3 
3 

436  9.31040 

0.68960  4.8933 

27 

34 

051  9.30213 

62 
61 

969  9.99109 

466  9.31104 

0.6880   860 

26 
25 

35 

20079  9-30275 

97963  9.99106 

20497  9.31168 

0.68832  4.8788 

3& 

108  9.30336 

958  999104 

3 

527  9.31233 

0.68767   716 

24 

37 

136  9-30398 

61 

952  9-99IOI 

557  9.31297 

0.68703   644 

23 

3« 

165  9-30459 

Ao 

946  9.99099 

3 
3 

588  9.31361 

0.68639   573 

22 

39 

193  9-30521 

6i 
61 
61 
61 
61 
61 
60 
61 
60 
61 
60 
61 
60 
60 

940  999096 

618  9-31425 

0.68575   501 

21 

40 

20222  9.30582 

97934  9-99093 

20648  9.31489 

0.68511  4.8430 

20 

41 

250  9.30643 

928  9-99091 

679  9-31552 

0.68448   359 

19 

42 

279  9-30704 

922  9.99088 

3 

709  9.31616 

0.68384   288 

18 

43 

307  9-30765 

916  9.99086 

3 
3 

739  9-31679 

0.68321   218 

17 

44 

336  9.30826 

910  9.99083 

770  9-31743 

0.68257   147 

lb 

45 

20364  9.30887 

97905  9.99080 

20800  9.31806 

0.68194  4.8077 

15 

46 

393  9-30947 

899  9.99078 

830  9.31870 

0.68130   007 

14 

47 

421  9.31008 

893  9-99075 

3 

861  9.31933 

0.68067  4.7937 

13 

48 

450  9.31068 

887  9.99072 

3 

891  931996 

0.68004   867 

12 

49 

478  9.31129 

881  9.99070 

3 

921  9.32059 

0.67941   798 

11 

50 

20507  9.31 189 

97875  9.99067 

20952  9.32122 

0.67878  4.7729 

10 

SI 

535  9-31250 

869  9.99064 

3 

982  9.32185 

0.67815   659 

9 

52 

563  9.31310 

863  9.99062 

3 
3 
2 

21013  9.32248 

0.67752   591 

8 

53 

592  9.31370 

60 
60 

857  9-99059 

043  9-32311 

0.67689   522 

7 

54 
55 

620  9-31430 

851  9-99056 

073  9-32373 

63 
62 

0.67627   453 

b 

20649  9-31490 

97845  9.99054 

21104  9.32436 

0.67564  4.7385 

5 

56 

(>77   9-31549 

60 

839  9-99051 

3 

134  9.32498 

t? 

0.67502   317 

4 

57 

706  9.31609 

833  9-99048 

3 

164  9.32561 

0.67439   249 

3 

58 

734  9-31669 

827  9.99046 

195  9.32623 

62 

0.67377   181 

2 

hi 

763  9.31728 

6^ 

821  9.99043 

3 

225  9.32685 

62 

0.67315   114 

I 

791  9.31788 

815  9.99040 

3 

256  9.32747 

0.67253   046 

0 

Nat.  Cos  Log.  d. 

Nat.  Sin  Log. 

d. 

Nat.  Cot  Log. 

c.d. 

Log.TanNat. 

/ 

78' 


12° 

t 

Nat.  Sin  Log.  d. 

Nat.  Cos  Log.  d. 

Nat.TanLog. 

c.d. 

Log.  Cot  Nat. 

0 

20791  9.31788 

59 

6n 

97815  9.99040 

21256  9.32747 

% 

0.67253  4.7046 

60 

I 

820  9.31847 

809  9-99038 

3 
3 
2 

286  9.32810 

0.67190  4.6979 

59 

2 

848  9.31907 

59 
59 
59 
59 
59 

59 
58 

59 
59 
58 
58 
59 
58 

58 
58 
58 
57 
58 
57 
58 
57 
57 
58 
57 
57 

% 
57 

803  9-99035 

316  9.32872 

61 

0.67128   912 

S8 

3 

877  9.31966 

797  9-99032 

347  9-32933 

6^^ 

0.67067   84s 

57 

4 

90s  9.32025 

791  9.99030 

3 

3 
2 

377  9-32995 

62 

50 

0.67005   779 

56 
55 

5 

20933  9-32084 

97784  9.99027 

21408  9.33057 

0.66943  4.6712 

6 

962  9.32143 

778  9-99024 

438  9-33119 

61 

0.66881   646 

S4 

7 

990  9.32202 

772  9.99022 

3 
3 
3 
2 

469  9-33180 

50 

0.66820   580 

53 

8 

21019  9.32261 

766  9.99019 

499  9-33242 

61 

0.66758   514 

52 

9 
10 

047  9-32319 

760  9.99016 

529  9-33303 
21560  9-33365 

62 
61 

0.66697   448 

51 

21076  9.32378 

97754  9-99013 

0.66635  4-6382 

50 

II 

104  9.32437 

748  9-9901 1 

3 
3 
3 
2 

3 

3 

3 
2 

590  9-33426 

61 

0.66574   317 

49 

12 

132  9.32495 

742  9-99008 

621  9.33487 

61 

0.66513   252 

48 

13 

161  9.32553 

735  999005 

651  9-33548 

61 

0.66452   187 

47 

14 

189  9.32612 

729  9.99002 

682  9.33609 

61 
61 

0.66391   122 

46 

15 

21218  9.32670 

97723  9-99000 

2 17 1 2  9.33670 

0.66330  4.6057 

45 

lb 

246  9.32728 

717  9-98997 

743  9-33731 

61 

0.66269  4.5993 

44 

17 

27s  9-32786 

711  9-98994 

773  933792 

0.66208   928 

43 

i8 

303  9.32844 

705  9.98991 

804  9-33853 

60 

0.66147  •  864 

42 

19 

331  9.32902 

698  9.98989 

3 
3 
3 
2 

834  9-33913 

61 

fin 

0.66087   800 

41 

20 

21360  9.32960 

97692  9.98986 

21864  9-33974 

0.66026  45736 

40 

21 

388  9.33018 

686  9.98983 

895  9-34034 

61 

0.65966   673 

39 

22 

417  9-33075 

680  9.98980 

925  9-3409$ 

f)n 

,0.6590$   609 

38 

23 

445  9-33133 

673  9-98978 

3 

3 

3 
2 

956  9-34155 

60 

0.65845   546 

37 

24 

25- 

474  9-33190 

667  9-98975 

986  9.34215 

61 

60 

0-65785   483 

36 

21502  9.33248 

97661  9.98972 

22017  9-34276 

0.65724  4.5420 

35 

26 

530  9-33305 

655  9.98969 

047  9-34336 

fin 

0.65664   357 

34 

27 

559  9-33362 

648  9.98967 

3 
3 
3 
3 

078  9-34396 

°°  0.65604    294 

33 

28 

587  9-33420 

642  9.98964 

108  9-34456 

60  \   0-65544    232 

32 

29 

6i6  9-33477 

636  9.98961 

139  9-34516 

60 

0.65484   169 

31 

21644  9-33534 

97630  9.98958 

22169  9-34576 

0.65424  4.5107 

30 

31 

672  9-33591 

623  9-98955 

200  9.34635 

^^  '  0.6536$    045 

29 

32 

701  9.33647 

3 
3 
3 
3 
3 

231  9-34695 

^°  0.6530$  4.4983 

28 

33 

729  9-33704 

611  9.98950 

261  9-34755  ''.  :X  0.65245   922 

27 

34 

758  9-33761 

57 
57 
56 
57 
56 
56 
57 

^i 

^l 

^l 
56 

56 

56 

55 

56 

55 

56 

55 

56 

55 

55 

55 

55 

55 

55 

55 

55 

604  9-98947 

292  9.34814 

2^  1  0.65186    860 

26 
25 

35 

21786  9.33818 

97598  9-98944 

22322  9.34874 

59 
59 
59 
6n 

0.65126  4-4799 

36 

814  9-33874 

592  9-98941 

353  9-34933 

0.65067   737 

24 

37 

843  9-33931 

585  9-98938 

383  9-34992 

0.65008   676 

23 

3B 

871  9-33987 

579  9-98936 

3 
3 
3 
3 
3 
2 

414  9-35051 

0.64949   615 

22 

39 

899  9.34043 

573  9-98933 

444  9-351" 

59 
59 
59 
59 
58 
59 
59 
58 
59 
58 
59 
58 
58 
58 
58 
58 

58 
57 

0.64889   555 

21 

40 

21928  9.34100 

97566  9.98930 

22475  9-35170 

0.64830  4.4494 

20 

41 

956  9.34156 

560  9-98927 

505  9.35229 

0.64771   434 

19 

42 

985  9.34212 

553  9-98924 

536  9-35288 

0.64712   373 

18 

43 

22013  9-34268 

547  9-98921 

567  9-35347 

0.64653   313 

17 

44 

041  9-34324 

541  9-98919 

3 
3 
3 
3 
3 
3 
3 

597  9-35405 

0.64595   253 

16 

15 

45 

22070  9.34380 

97534  9-98916 

22628  9.35464 

0.64536  4-4194 

46 

098  9-34436 

528  9.98913 

658  9-35523 

0.64477   134 

14 

47 

126  9.34491 

521  9.98910 

689  9-35581 

0.64419   075 

13 

48 

155  9-34547 

515  9-98907 

719  9-35640 

0.64360   015 

12 

49 

183  9.34602 

508  9.98904 

750  9-35698 

0.64302  4.3956 

II 

50 

22212  9.34658 

97502  9.98901 

22781  9.35757 

0.64243  4-3897 

10 

SI 

240  9-34713 

496  9.98898 

811  9.35815 

0.64x85   838 

9 

S2 

268  9.34769 

489  9-98896 

3 
3 
3 

842  9-35873 

0.64127   779 

8 

S3 

297  9-34824 

483  9.98893 

0.64069   721 

7 

54 

325  9-34879 

476  9.98890 

903  9.35989 

0.64011   662 

b 
^5 

55 

22353  9-34934 

97470  9.98887 

22934  9-36047 

0.63953  4-3604 

S6 

382  9.34989 

463  9.98884 

3 
3 
3 
3 
3 

964  9.36105 

0.63895   546 

4 

S7 

410  9.35044 

457  9-98881 

995  9-36163 

0.63837   488 

3 

,S8 

438  9-35099 

450  9-98878 

23026  9.36221 

0.63779   430 

2 

S9 

467  9-35154 

444  9-98875 

056  9-36279 

0.63721   372 

I 

60 

495  9-35209 

437  9-98872 

087  9-36336 

0.63664   315 

0 

Nat.  Cos  Log.  d. 

Nat.  Sin  Log.  d. 

Nat.  Cot  Log. 

c.d. 

Log.TanNat. 

/ 

77' 


Nat.  Sin  Log.  d. 


13^ 

Nat.  Cos  Log.  d. 


Nat.TanLog.  c.d 


Log.  Cot  Nat. 


22495 
523 
552 
580 
608 


9-35209 
935263 
935318 
9-35373 
9-35427 


22637 
665 

693 
722 

750 


9-35481 
9-35536 
9-35590 
935644 
9-35698 


22778 
807 

835 
863 
892 


9-35752 
9.35806 
9-35860 

9-35914 
9-35968 


22920 
948 

977 
23005 

033 


9.36022 

9-36075 
9.36129 
9.36182 
9-36236 


23062 
090 
118 
146 
175 


9.36289 
9-36342 
9-36395 
9-36449 
9.36502 


23203 
231 
260 
288 
316 


9-36555 
9.36608 
9.36660 
9-36713 
9-36766 


23345 
373 
401 
429 
458 


9.36819 
9-36871 
9-36924 
9.36976 
9.37028 


23486 
514 
542 
571 
599 


9-37081 

9-37133 
9-37185 
9-37237 
9-37289 


23627 
656 
684 
712 
740 


9-37341 
9-37393 
9-37445 
9-37497 
9-37549 


23769 

797 
825 

853 
882 


9-37600 
9-37652 
937703 
9-37755 
9.37806 


23910 
938 
966 

995 
24023 


9-37858 
937909 
9-37960 
9.3801 1 
9.38062 


24051 
079 
108 
136 
164 
192 


9-381 13 
9.38164 
9.38215 
9.38266 

9-38317 
9-38368 


97437 
430 
424 
417 
411 


9.98872 
9.98869 
9.98867 
9.98864 
9.98861 


97404 
398 
391 
384 
378 


9-98858 

9-98855 
9.98852 
9.98849 
9.98846 


97371 
365 
358 
351 
345 


9-98843 
9.98840 
9.98837 

9-98834 
9.98831 


97338 
331 
325 
318 
3" 


9.98828 
9.98825 
9.98822 
9.98819 
9.98816 


97304 
298 
291 
284 
278 


9-98813 
9.98810 
9.98807 
9.98804 
9.98801 


97271 
264 
257 
251 
244 


9.98798 

9-98795 
9.98792 
9.98789 
9.98786 


97237 
230 
223 
217 
210 


9-98783 
9.98780 
9.98777 

9-98774 
9.98771 


97203 
196 
189 
182 
176 


9.98768 

9-98765 
9.98762 

9-98759 
9-98756 


97169 
162 

15s 
148 
141 


9-98753 
9.98750 
9-98746 
9-98743 
9-98740 


97134 
127 
120 

"3 
106 


9-98737 
9-98734 
9.98731 
9.98728 
9-98725 


97100 

093 
086 
079 
072 


9.98722 
9.98719 
9.98715 
9.98712 
9-98709 


97065 
058 

051 
044 

037 
030 


9.98706 
9.98703 
9.98700 
9.98697 

9-98694 
9.98690 


23087 
117 
148 
179 
209 


9-36336 
9-36394 
9-36452 
936509 
9.36566 


23240 
271 
301 
332 
363 


9.36624 
9.36681 
9-36738 

9-36795 
9.36852 


23393 
424 
455 
485 
516 


9-36909 
9-36966 
937023 
9-37080 
9-37137 


23547 
578 
608 

639 
670 


9-37193 
9-37250 
9-37306 
9-37363 
9-37419 


23700 

731 
762 

793 
823 


9-37476 
937532 
937588 
9-37644 
9-37700 


23854 
885 
916 
946 
977 


9-37756 
9-37812 
9.37868 

9-37924 
9.37980 


24008 

039 
069 
100 
131 


9-38035 
9.38091 
9.38147 
9.38202 
9-38257 


24162 

193 
223 

254 
285 


9-.38313 
9.38368 

9-38423 
9-38479 
9-38534 


24316 
347 
377 
408 

439 


9.38589 
9-38644 
9-38699 
9-38754 
9.38808 


24470 
501 
532 

562 

593 


9-38863 
9.38918 
9.38972 
9.39027 
9-39082 


24624 

655 
686 
717 
747 


9-39136 
9.39190 
9-39245 
9-39299 
9-39353 


24778 
809 
840 
871 
902 
933 


9-39407 
9.39461 

9-39515 
9-39569 
9.39623 
9.39677 


0.63664 
0.63606 
0.63548 
0.63491 
0-63434 


4-331S 
257 
200 

143 
086 


0.63376 
0.63319 
0.63262 
0.63205 
0.63148 


4.3029 

4.2972 

916 

859 
803 


0.63091 
0.63034 
0.62977 
0.62920 
0.62863 


4.2747 
691 

635 
580 

524 


0.62807 
0.62750 
0.62694 
0.62637 
0.62581 


4.2468 
413 
358 
303 
248 


0.62524 
0.62468 
0.62412 
0.62356 
0.62300 


4.2193 

139 

084 

030 

4.1976 


0.62244 
0.62188 
0,62132 
0.62076 
0.62020 


4.1922 
868 
814 
760 
706 


0.61965 
0.61909 
0.61853 
0.61798 
0.61743 


4-1653 
600 
547 
493 
441 


0.61687 
0.61632 
0.61577 
0.61521 
0.61466 


4.1388 

335 
282 
230 
178 


0.61411 
0.61356 
0.61301 
0.61246 
0.61192 


4.1126 
074 
022 

4.0970 
918 


0.61 137 
0.61082 
0.61028 
0.60973 
0.60918 


4.0867 

81S 
764 

713 
662 


0.60864 
0.60810 
0.60755 
0.60701 
0.60647 


4.061 1 
560 
509 
459 
408 


0.60593 
0.60539 
o.6o48g 
0.60431 
0.60377 
0.60323 


4.0358 
308 

257 
207 

158 
108 


Nat.  Cos  Log.  d. 


Nat.  Sin  Log.  d. 

76^ 


Nat.  Cot  Log, 


d.  Log. Tan  Nat 


w 


r 

Nat.  Sin  Log.  d. 

Nat.  Cos  Log.  d. 

Nat.TanLog. 

c.d. 

Log.  Cot  Nat. 

0 

24192  9.38368 

50 
51 
50 
51 
50 
50 
51 
50 
50 
50 
50 
50 
50 
50 
50 

97030  9.98690 

3 
3 

24933  9-39677 

54 

0.60323  4.0108 

60 

I 

220  9.38418 

023  9.98687 

964  9-39731 

0.60269   058 

59 

2 

249  9-38469 

015  9.98684 

995  9-39785 

0.60215   009 

58 

3 

277  9-38519 

008  9.98681 

3 

25026  9.39838 

53 

0.60162  3.9959 

57 

4 
5 

305  9-38570 

001  9.98678 

3 
3 

056  9-39892 

54 
53 

0.60108   910 

56 
56 

24333  9-38620 

96994  9.98675 

25087  9-39945 

0.60055  3.9861 

b 

362  9.38670 

987  9-98671 

118  9.39999 

54 

0.60001   812 

54 

7 

390  9.38721 

980  9.98668 

149  9-40052 

53 

0.59948   763 

53 

8 

418  9.38771 

973  9-98665 

3 
3 

180  9.40106 

54 

0.59894   714 

52 

9 

446  9.38821 

966  9.98662 

211  9.40159 

53 
53 

0.59841   665 

51 
60 

10 

24474  9-38871 

96959  9-98659 

25242  9.40212 

0-59788  3-9617 

II 

503  9.38921 

952  9-98656 

3 
4 
3 

273  9.40266 

54 
53 
53 

0-59734   568 

49 

12 

531  9.38971 

945  998652 

304  9.40319 

0.59681   520 

48 

13 

559  9-39021 

937  9-98649 

335  9-40372 

0.59628   471 

47 

14 

587  9-39071 

930  9.98646 

3 
3 

366  9.40425 

53 
53 

0-59575   423 

46 
46 

16 

24615  9.39121 

96923  9.98643 

25397  9-40478 

0.59522  3-9375 

I6 

644  9.39170 

916  9.98640 

3 

428  9.40531 

53 

0.59469   327 

44 

17 

672  9.39220 

50 

909  9.98636 

459  9-40584 

53 

0.59416   279 

43 

i8 

700  9.39270 

50 

902  9.98633 

3 

490  9.40636 

52 

0.59364   232 

42 

19 

728  9.39319 

50 

894  9.98630 

3 
3 

521  9.40689 

53 
53 

0.5931 1   184 

41 
40 

20 

24756  9-39369 

96887  9.98627 

25552  9.40742 

0-59258  3-9136 

21 

784  9.39418 

49 

880  9.98623 

4 

583  9.40795 

53 

0.59205   089 

39 

22 

813  9-39467 

49 

873  9.98620 

3 

614  9.40847 

52 

0.59153   042 

38 

23 

841  9-39517 

50 

866  9.98617 

3 

645  9.40900 

53 

0.59100  3.8995 

37 

24 

25 

869  9-39566 

49 
49 

858  9.98614 

3 

4 

676  9.40952 

52 
53 

0.59048   947 

36 

24897  9-39615 

96851  9.98610 

25707  9.41005 

0.58995  3.8900 

35 

26 

925  9.39664 

844  9.98607 

3 

738-  9-41057 

52 

0-58943   854 

34 

27 

954  9-39713 

837  9.98604 

3 

769  9.41109 

52 

0.58891   807 

33 

28 

982  9.39762 

829  9.98601 

3 

800  9.41161 

52 

0.58839   760 

32 

29 

25010  9.39811 

49 

822  9.98597 

4 
3 

831  9.41214 

53 
52 

0.58786   714 

31 
30 

30 

25038  9-39860 

96815  9-98594 

25862  9.41266 

0.58734  3.8667 

31 

066  9.39909 

49 

807  9.98591 

3 

893  9.41318 

52 

0.58682   621 

29 

32 

094  9-39958 

49 

48 

800  9.98588 

3 

924  9.41370 

52 

0.58630   575 

28 

33 

122  9.40006 

793  9-98584 

4 

955  9-41422 

52 

0.58578   528 

27 

34 

151  9.40055 

49 
48 

49 
48 

786  9.98581 

3 
3 

986  9.41474 

52 
52 
52 
51 

0.58526   482 
0.58474  3-8436 

26 
26 

35 

25179  9.40x03 

96778  9-98578 

26017  9.41526 

3^ 

207  9.40152 

77^   9-98574 

4 

048  9.41578 

0.58422   391 

24 

37 

235  9.40200 

764  9.98571 

3 

079  9.41629 

0.58371   345 

23 

3« 

263  9.40249 

48 
49 
48 
48 
48 
48 
48 

48 

47 
48 
48 

756  9-98568 

3 

no  9.41681 

52 

0.58319   299 

22 

39 
40 

291  9.40297 

749  9-98565 

3 

4 

141  9.41733 

52 
51 
52 
51 
52 
51 
51 
52 
51 
51 
51 
51 
51 
51 
51 
51 

% 

50 
51 
51 
50 

0.58267   254 

21 

25320  9-40346 

96742  9.98561 

26172  941784 

0.58216  3.8208 

20 

41 

348  9.40394 

734  9-98558 

3 

203  941836 

0.58164   163 

19 

42 

376  9.40442 

727  9-98555 

3 

235  941887 

O.58113   118 

18 

43 

404  9.40490 

719  9-98551 

4 

266  9.41939 

0.58061   073 

17 

44 

432  9.40538 

712  9.98548 

3 
3 

297  9-41990 

0.58010   028 

lb 

45 

25460  9.40586 

96705  9.98545 

26328  9.42041 

0-57959  3-7983 

16 

46 

488  9.40634 

697  9-98541 

4 

359  9-42093 

0-57907   938 

14 

47 

516  9.40682 

690  9.98538 

3 

390  942144 

0.57856   893 

13 

48 

545  9-40730 

682  9.98535 

3 

421  9.42195 

0.57805   848 

12 

49 

573  9-40778 

675  9-98531 

4 
3 

452  942246 

0-57754   804 

II 
To 

50 

25601  9.40825 

96667  9.98528 

26483  9.42297 

0-57703  3-7760 

SI 

629  9.40873 

660  9.98525 

3 

515  9.42348 

0.57652   715 

9 

52 

657  9-40921 

653  9-98521 

546  9-42399 

0.57601   671 

8 

53 

685  9.40968 

48 
47 
48 

645  9-98518 

3 

577  9-42450 

0-57550   627 

7 

54 
55~ 

713  9.41016 

638  9-98515 

3 

4' 

608  9.42501 

0.57499   583 

b 

25741  9.41063 

96630  9.98511 

26639  9-42552 

0-57448  3-7539 

5 

5^ 

769  941111 

623  9.98508 

3 

670  9.42603 

0.57397   495 

4 

57 

798  9-4"58 

615  9.98505 

3 

701  9-42653 

0.57347   451 

3 

5« 

826  9.41205 

47 

608  9.98501 

4 

733  9-42704 

0.57296   408 

2 

18 

854  9.41252 

48 

600  9.98498 

3 

764  9.42755 

0-57245   364 

1 

882  9.41300 

593  998494 

4 

795  942805 

0.5719S   321 

U 

Nat.  Cos  Log.  d. 

Nat.  Sin  Log.  d. 

Nat.  Cot  Log. 

c.d. 

Log. Tan  Nat. 

t 

w 


15: 

Nat.  Sin  Log.  d.  Nat.  Cos  Log.  d.  Nat.TanLog.  c.d.lLog.CotNat 


25882 
910 
938 
966 
994 


941300 
941347 
941394 
9.41441 
9.41488 


26022 
050 
079 
107 
135 


941535 
941582 
941628 
9.41675 
941722 


26163 
191 
219 
247 

275 


9.41768 
941815 
941861 
9.41908 
941954 


26303 
331 
359 
387 
415 


9.42001 
9.42047 
9.42093 
942140 
942186 


26443 

471 
500 
528 
556 


942232 
942278 
942324 
942370 
942416 


26584 
612 
640 
668 
696 


942461 
942507 
942553 
9.42599 
9.42644 


26724 
752 
780 
808 
836 


942690 

9.42735 
9.42781 
9.42826 
942872 


26864 
892 
920 
948 
976 


942917 
9.42962 
943008 

9.43053 
943098 


27004 
032 
060 
088 
116 


943143 
9.43188 

943233 
9.43278 

943323 


27144 
172 
200 
228 
2c;6 


943367 
943412 

943457 
943502 
943546 


27284 
312 
340 
368 
396 


943591 
943635 
943680 

9.43724 
943769 


27424 
452 
480 
508 
536 
564 


9-43813 
943857 
9.43901 
943946 
943990 
944034 


96593 

585 
578 
570 
562 


9.98494 
9.98491 
9.98488 
9.98484 
9.98481 


96555 
547 
540 
532 
524 


9.98477 
9.98474 
9.98471 
9.98467 
9.98464 


96517 
509 
502 

494 
486 


9.98460 
998457 
9-984$3 
9.98450 
9.98447 


96479 
471 
463 
456 
448 


998443 
9.98440 
9.98436 

9.98433 
9.98429 


96440 
433 
425 
417 
410 


9^98426 
9.98422 
9.98419 

9.98415 
9.98412 


96402 
394 
386 
379 
371 


9.98409 

9.98405 
9.98402 

9.98398 
9.98395 


96363 
355 
347 
340 
332 


9.98391 
998388 
9.98384 
9.98381 
9.98377 


96324 
316 
308 
301 
293 


9.98373 
9.98370 
9.98366 
998363 
9.98359 


96285 

277 
269 
261 
253 


9.98356 
998352 
9.98349 

9.98345 
9.98342 


96246 
238 
230 
222 
214 


9.98338 
9.98334 
9-98331 
9.98327 
9.98324 


96206 
198 
190 
182 
174 


9.98320 
9.98317 

9.98313 
9.98309 
9.98306 


96166 
158 
150 
142 

134 
126 


9.98302 
9.98299 

9.98295 
9.98291 
9.98288 
9.98284 


26795 
826 

857 
888 
920 


9.42805 
942856 
942906 
9.42957 
9.43007 


26951 

943057 

982 

9.43108 

27013 

943158 

044 

9.43208 

076  9.43258 

27107  943308 

138  9.43358 

169 

943408 

201 

943458 

232 

943508 

27263  9.43558 

294 

9.43607 

326  943657 

357 

943707 

388  943756 

27419 

9.43806 

451 

9.43855 

482  9.43905 

513 

943954 

545 

9.44004 

27576  944053 

607 

9.44102 

638 

9.44151 

670 

944201 

701 

9.44250 

27732  9.44299 

764  944348 

795 

944397 

826 

9.44446 

858  944495 

27889 

944544 

921 

9.44592 

952 

944641 

983  9.44690 

28015  944738 

28046  9.44787 

077 

944836 

109 

9.44884 

140 

9.44933 

172 

9.44981 

28203  9.45029 

234 

9.45078 

266 

9.45126 

297  945174 

329 

9.45222 

28360  945271 

391 

9.45319 

423 

945367 

454 

9.45415 

486  945463 

28517 

549 
580 
612 
643 
675 


9455" 
9.45559 
9.45606 

9.45654 
945702 
945750 


0.57195 
0.57144 

0.57094 
0.57043 
0.56993 


3-7321 
277 

234 
191 
148 


0.56943 
0.56892 
0.56842 
0.56792 
0.56742 


3.7105 
062 
019 

3.6976 
933 


0.56692 
0.56642 
0.56592 
0.56542 
0.56492 


3.6891 
848 
806 
764 
722 


0.56442 
0.56393 
0.56343 
0.56293 
0.56244 


3.6680 
638 
596 
554 
512 


0.56194 
0.5614$ 
0.56095 
0.56046 
0.55996 


3.6470 
429 
387 
346 
30s 


0.55947 
0.55898 

0.55849 
0.55799 
0-55750 


3.6264 
222 
181 
140 
100 


0.55701 
0.55652 
0.55603 

0.55554 
0.55505 


3.6059 
018 

3-5978 
937 
897 


0.55456 
0.55408 
0-55359 
0.55310 
0.55262 


3.5856 
816 
776 
736 
696 


0.55213 
0.55164 
0.551 16 
0.55067 
0.55019 


3.5656 
616 
576 
536 
497 


0.54971 
0.54922 

0.54874 
0.54826 
0.54778 


3-5457 
418 

379 
339 
300 


0.54729 
0.54681 

0.54633 
0.54585 
0.54537 


3.5261 
222 

183 
144 

105 


0.54489 
0.54441 
0.54394 
0.54346 
0.54298 
0.54250 


3-5067 

028 

3.4989 

951 
912 

874 


Nat.  Cos  Log.  d.  Nat.  Sin  Log.  d.  Nat.Cot  Log.  c.d.  Log.TanNat 


74' 


16' 

'   Nat.  Sin  Log.  d.  Nat.  Cos  Log.  d. 


Nat.TanLog.  c.d 


Log.  Cot  Nat, 


27564 
592 
620 
648 
676 


9.44034 
9.44078 
9.44122 
9.44166 
9.44210 


27704 
731 

759 
787 

815 


9.44253 
9.44297 

9-44341 
944385 
9.44428 


27843 
871 
899 
927 
955 


9.44472 
9.44516 

9-44559 
9.44602 
9.44646 


27983 
2801 1 

039 
067 

095 


9.44689 

9-44733 
9.44776 
9.44819 
9.44862 


28123 

150 
178 
206 
234 


9.44905 
9.44948 
9.44992 
945035 
9-45077 


28262 
290 
318 
346 
374 


9.45120 
9-45163 
9.45206 
9.45249 
9.45292 


28402 
429 
457 
485 
513 


9-45334 
9-45377 
9-45419 
945462 

9-45504 


28541 
569 
597 
625 
652 


9-45547 
9-45589 
9-45632 
9-45674 
9-45716 


28680 
708 
736 
764 
792 


9-45758 
9-45801 
9-45843 
9-45885 
9-45927 


847 
875 
903 
931 


9.45969 
9.4601 1 
9-46053 
9-46095 
9.46136 


28959 
987 

29015 
042 
070 


9.46178 
9.46220 
9.46262 
9-46303 
9-46345 


29098 
126 

154 
182 
209 
237 


9.46386 
9.46428 
9.46469 
9-46511 
9-46552 
9-46594 


96126 
118 
no 
102 
094 


9.98284 
9.98281 

9-98277 
9.98273 
9.98270 


96086 
078 
070 
062 
054 


9.98266 
9.98262 
9-98259 
9-98255 
9.98251 


96046 

037 
029 
021 
013 


9.98248 
9.98244 
9.98240 
9.98237 
9-98233 


96005 

95997 
989 
981 
972 


9.98229 
9.98226 
9.98222 
9.98218 
9.98215 


95964 
956 
948 
940 
931 


9.9821 1 
9.98207 
9.98204 
9.98200 
9.98196 


95923 
915 
907 
898 
890 


9.98192 
9.98189 
9.98185 
9.98181 
9.98177 


95882 

874 
865 

857 


9.98174 
9.98170 
9.98166 
9.98162 
9.98159 


95841 
832 
824 
816 
807 


9-98155 
9.98151 
9.98147 
9.98144 
9.98140 


95799 
791 
782 

774 
766 


9-98136 
9.98132 
9.98129 
9.98125 
9.98121 


95757 
749 
740 
732 
724 


9.981 17 

9-98113 
9.981 10 
9.98106 
9.98102 


95715 
707 
698 
690 
681 


9.98098 
9.98094 
9.98090 
9.98087 
9-98083 


95673 
664 
656 
647 

639 
630 


9.98079 

9-98075 
9.98071 
9.98067 
9-98063 
9.98060 


28675 
706 
738 
769 
801 


9-45750 
9-45797 
9-45845 
9.45892 

9-45940 


28832 
864 

895 
927 

958 


9-45987 
9-46035 
9.46082 
9.46130 
9.46177 


28990 
29021 

053 
084 
116 


9.46224 
9.46271 
9.46319 
9-46366 
9.46413 


29147 
179 
210 
242 
274 


9.46460 
9-46507 
9-46554 
9.46601 
9.46648 


29305 
337 
368 
400 
432 


•9.46694 
9.46741 
946788 
9-46835 
9.46881 


29463 

495 
526 

558 
590 


9.46928 

9-46975 
9.47021 
9.47068 
9.47114 


29621 

653 
685 
716 
748 


9.47160 
9.47207 

9-47253 
9.47299 

9-47346 


29780 
811 
843 
875 
906 


9-47392 
9-47438 
9-47484 
9-47530 
9-47576 


29938 

970 

30001 

033 
065 


9.47622 
9.47668 
9.47714 
9-47760 
9.47806 


30097 
128 
160 
192 
224 


947852 
9-47897 
9-47943 
947989 
9-48035 


30255 

287 

319 
351 
382 


9.48080 
9.48126 
9.48171 
9.48217 
9.48262 


30414 
446 
478 
509 
541 
573 


9.48307 

9-48353 
9.48398 

9-48443 
9.48489 

948534 


0.54250 
0.54203 
0.54155 
0.54108 
0.54060 


3-4874 
836 
798 
760 
722 


0.54013 

0-53965 
0.53918 
0.53870 
0.53823 


3.4684 
646 
608 
570 
533 


0.53776 
0.53729 
0.53681 
0-53634 
0-53587 


3-4495 
458 
420 
383 
346 


0.53540 
0.53493 
0.53446 
0.53399 
0.53352 


3-4308 
271 

234 
197 
160 


0.53306 

0.53259 
0.53212 
0.53165 
0.53119 


3.4124 
087 
050 
014 

3.3977 


0.53072 
0.53025 
0.52979 
0.52932 
0.52886 


3-3941 
904 
868 
832 
796 


0.52840 
0-52793 
0-52747 
0.52701 
0.52654 


3-3759 
723 
687 
652 
616 


0.52608 
0.52562 
0.52516 
0.52470 
0.52424 


3-3580 
544 
509 
473 
438 


0.52378 
0.52332 
0.52286 
0.52240 
0-52194 


3-3402 
367 
332 
297 
261 


0.52148 
0.52103 
0.52057 
0.5201 1 
0.51965 


3-3226 
191 
156 
122 
087 


0.51920 

0.51874 
0.51829 

0.51783 
0.51738 


3-3052 
017 

3-2983 
948 
914 


0.51693 
0.51647 
0.51602 
0-51557 

0.51466 


3-2879 
845 
811 

m 
743 
709 


Nat.  Cos  Log.  d. 


Nat.  Sin  Log.  d. 

73^ 


Nat.  Cot  Log 


c.d.  Log. Tan  Nat 


17° 


Nat.  Sin  Log.  d.  Nat.  Cos  Log.  d.  Nat.TailLog.  c.d.lLog.  Cot  Nat. 


29237 
265 

293 
321 

348 


9.46594 

946635 
9.46676 
9.46717 
9.46758 


29376 
404 
432 
460 
487 


9.46800 
946841 
946882 
9.46923 
9.46964 


10  '  29515 


543 
571 
599 
626 


9.4700$ 

947045 
9.47086 
947127 
9.47168 


29654 
682 
710 
737 
765 


9.47209 
9.47249 
9.47290 
947330 
947371 


29793 
821 
849 
876 
904 


9.4741 1 
947452 
9.47492 

947533 
947573 


29932 
960 
987 

30015 
043 


947613 

947654 
9.47694 

947734 
9.47774 


30071 
098 
126 

154 

182 


9.47814 

947854 
9.47894 

947934 
947974 


30209 
237 
265 
292 
320 


948014 
9.48054 
9.48094 

948133 
9.48173 


30348 
376 
403 
431 
459 

30486 
514 
542 
570 
597 


9.48213 
948252 
9.48292 
948332 
948371 
94841 1 
9.48450 
9.48490 
9.48529 
9.48568 


30625 

653 
680 
708 
736 


9.48607 
9.48647 
9.48686 
9.48725 
9.48764 


30763 
791 
819 
846 

874 
902 


948803 
9.48842 
948881 
9.48920 
9.48959 
9.48998 


95630 
622 
613 
605 
596 


9.98060 
9.98056 
9.98052 
9.98048 
9.98044 


95588 
579 
571 
562 
554 


9.98040 
9.98036 
9.98032 
9.98029 
9.98025 


95545 
536 
528 
519 
5" 


9.98021 
9.98017 
9.98013 
9.98009 
9-98005 


95502 
493 
485 
476 
467 


9.98001 
9-97997 
9-97993 
9.97989 
9-97986 


95459 
450 
441 

433 
424 


9.97982 
9.97978 

9-97974 
9.97970 
9.97966 


95415 
407 
398 
389 
380 


9.97962 
9-97958 
9-97954 
9-97950 
9-97946 


95372 
363 
354 
345 
337 


9-97942 
997938 
9-97934 
9-97930 
9-97926 


95328 
319 
310 
301 
293 


9.97922 
9.97918 
9.97914 
9.97910 
9-97906 


95284 

275 
266 

257 
248 


9.97902 
9.97898 
9.97894 
9.97890 
9.97886 


95240 
231 
222 
213 
204 


9.97882 
9-97878 
9.97874 
9-97870 
997866 


95195 
186 
177 
168 
159 


9.97861 
997857 
9-97853 
9-97849 
9-97845 


95150 
142 

133 
124 

"5 
106 


9.97841 
9-97837 
9-97833 
9-97829 
9.97825 
9.97821 


30573 
605 

637 
669 
700 


9-48534 
948579 
9.48624 
9.48669 
9.48714 


30732 

9-48759 

764  9-48804 

796  948849 

828 

9.48894 

860 

9.48939 

30891 

948984 

923 

9.49029 

955 

9-49073 

987  9491 18 

31019 

9.49163 

31051 

9.49207 

083  949252 

"5 

9.49296 

147 

9-49341 

178  9-49385 

31210 

9-49430 

242 

9-49474 

274 

9-49519 

306  949563 

338  949607 

31.370 

9-49652 

402 

949696 

434 

9-49740 

466  9.49784 

498  9.49828 

31530 

9.49872 

562 

9.49916 

594 

949960 

626 

9.50004 

658  9.50048 

31690  9.50092 

722 

9-50136 

754 

9.50180 

786  9.50223 

818 

9.50267 

31850  9.5031 1 

882 

9-50355 

914 

9-50398 

946  9.50442 

978  9-50485 

32010  9.50529 

042 

9-50572 

074 

9.50616 

106 

9-50659 

139 

950703 

32171 

9-50746 

203 

9.50789 

235 

950833 

267  9.50876 

299 

9.50919 

32331 
363 
396 
428 
460 
492 


9-50962 
9.51005 
9.51048 
9.51092 

9.5II35 
9.51 178 


0.51466 
0.5I42I 
0-51376 
0-5I33I 

0.51286 


3-2709 
675 

641 
607 
573 


0.51241 
0.51 196 

0.51151 
0.51106 
0.51061 


3-2539 
506 
472 
438 
405 


0.51016 
0.50971 
0.50927 
0.50882 
0.50837 


3-2371 
338 
305 
272 
238 


0-50793 
0-50748 
0.50704 
0.50659 
0.50615 


3-2205 
172 

139 
106 

073 


0.50570 
0.50526 
0.50481 
0-50437 
0.50393 


3-2041 
008 

3-1975 
943 
910 


0.50348 
0.50304 
0.50260 
0.50216 
0.50172 


3-1878 
845 
813 
780 
748 


0.50128 
0.50084 
0.50040 
0.49996 
049952 


3.1716 
684 
652 
620 


0.49908 
0.49864 
0.49820 
049777 
049733 


3-1556 
524 
492 
460 
429 


049689 
0.49645 
0.49602 
0.49558 
0.49515 


3-1397 
366 

334 
303 
271 


0.49471 
0.49428 
0.49384 
0.49341 
049297 


3.1240 
209 
178 
146 
"5 


0-49254 
0.4921 1 
0.49167 
0.49124 
0.49081 


3.1084 

053 

022 

3.0991 

961 


0.49038 
04899S 
0.48952 
0.48908 
0.48865 
048822 


3.0930 

899 
868 
838 
807 

m 


Nat.  Cos  Log.  d. 


Nat.  Sin  Log,  d.  |Nat.  Cot  Log,  c.d.  Log.TanNat. 

72" 


18 

D 

r 

Nat.  Sin  Log.  d. 

Nat.  Cos  Log.  d. 

Nat.TanLog. 

c.d. 

Log.  Cot  Nat. 

0 

30902  9.48998 

39 

39 

39- 

38 

39 

39 

38 

39 

95106  9.97821 

4 
5 

32492  9.5II78 

43 

0.48822  3.0777 

60 

I 

929  9.49037 

097  9.97817 

524  9,51221 

0.48779   746 

59 

2 

957  949076 

088  9.97812 

556  9.51264 

43 

0.48736   716 

58 

3 

985  9.49115 

079  9.97808 

4 
4 
4 

588  9.51306 

42 
43 
43 

0.48694   686 

57 

4 
5 

31012  9.49153 

070  9.97804 

621  9.51349 

0.48651   655 

56 
55 

31040  9.49192 

95061  9,97800 

32653  9.51392 

0.48608  3.0625 

6 

068  9.49231 

052  9-97796 

4 

685  9.51435 

43 

0.48565   595 

.54 

7 

095  9.49269 

043  9-97792 

717  9.51478 

0.48522   565 

53 

8 

033  9-97788 

.  749  9.51520 

0.48480   535 

52 

9 

151  949347 

38 
38 

024  9.97784 

5 
4 
4 
4 

782  9.51563 

43 
42 
43 
43 

0.48437   505 

51 
50 

10 

3 1 178  949385 

95015  9-97779 

32814  9.51606 

0.48394  3.0475 

II 

206  9.49424 

006  9.97775 

846  9.51648 

0.48352   445 

49 

12 

233  949462 

94997  9-97771 

878  9.51691 

0.483O9   415 

48 

13 

261  9.49500 

988  9-97767 

911  9.51734 

0.48266   385 

47 

14 

289  9.49539 

39 
38 
38 
39 
38 
38 
38 
38 
38 
38 
38 
38 
38 

3? 
38 

38 

38 

979  9-97763 

4 
4 
5 
4 
4 
4 
4 
4 
5 

943  9.51776 

43 
42 
42 
43 
42 
43 
42 
42 

0.48224   356 

4b 
45 

16 

31316  9.49577 

94970  9.97759 

32975  9.51819 

0.48181  3.0326 

lb 

344  949615 

961  9-977$4 

33007  9.51861 

0.48139   296 

44 

17 

372  9.49654 

952  997750 

040  9.51903 

0.48097   267 

43 

lb 

399  949692 

943  9-97746 

072  9.51946 

0.48054   237 

42 

19 
20 

427  949730 
31454  949768 

933  9-97742 

104  9.51988 

0.48012   208 

41 
40 

94924  9.97738 

33136  9.52031 

0.47969  3.0178 

21 

482  9.49806 

915  9-97734 

169  9.52073 

0.47927   149 

39 

22 

510  9.49844 

906  9.97729 

201  9.52115 

0,47885   120 

38 

23 

537  949882 

897  9.97725 

233  9.52157 

43 
42 

42 
42 
42 
42 
42 

0.47843   090 

37 

24 

565  9.49920 

888  9.97721 

4 

266  9.52200 

0.47800   061 

3^ 
35 

25 

31593  949958 

94878  9-97717 

33298  9..52242 

0.47758  3.0032 

2b 

620  9.49996 

869  9.97713 

4 
5 
4 
4 
4 
5 

0.47716   003 

34 

27 

648  9.50034 

860  9.97708 

363  9.52326 

0.47674  2.9974 

33 

28 

675  9.50072 

851  9.97704 

395  9.52368 

047632   945 

32 

29 

703  9.501 10 

842  9.97700 

427  9.52410 

0.47590   916 

31 
30 

30 

31730  9.50148 

94832  9.97696 

33460  9.52452 

0.47548  2.9887 

31 

758  9-50185 

38 

823  9-97691 

492  9.52494 

42 
42 

0.47506   858 

29 

32 

786  9.50223 

814  9-97687 

524  9.52536 

0.47464   829 

28 

33 

813  9.50261 

805  9.97683 

4 

557  9.52578 

0.47422   800 

27 

34 
35 

841  9.50298 

37 
38 
38 
37 
38 

795  997679 

5 

589  9,52620 

41 
42 
42 
42 
42 
41 
42 
41 
42 

0.47380   772 

2b 

25 

31868  9.50336 

94786  9.97674 

33621  9.52661 

0.47339  2.9743 

36 

896  9-50374 

777   997670 

4 

654  9.52703 

0.47297   714 

24 

37 

923  9-50411 

768  9.97666 

686  9.52745 

0.47255   686 

23 

3H 

951  9-50449 

758  9-97662 

5 
4 

718  9.52787 

0.47213   657 

22 

39 
40 

979  9-50486 

37 
37 
38 

749  9-97657 

751  9.52829 

0.47171   629 

21 
20 

32006  9.50523 

94740  9-97653 

33783  9.52870 

0.47130  2,9600 

41 

034  9-50561 

730  9.97649 

816  9,52912 

0.47088   572 

19 

42 

061  9.50598 

37 

721  9.97645 

4 

5 

848  9.52953 

0.47047   544 

i8 

43 

089  9.50635 

^8 
37 
37 

712  9.97640 

881  9.52995 

o.470og   515 

17 

44 

116  9-50673 

702  9.97636 

4 
4 
4 
5 
4 

913  9.53037 

4- 
41 
42 
41 
41 

0.46963   487 

lb 
15 

45 

32144  9.50710 

94693  9.97632 

33945  9.53078 

0.46922  2,9459 

4b 

171  9-50747 

684  9.97628 

978  9.53120 

0.46880   431 

14 

47 

199  9-50784 

37 
37 

674  9.97623 

34010  9.53161 

0.46839   403 

13 

48 

227  9.50821 

665  9.97619 

043  9.53202 

0.46798   375 

12 

49 
50 

254  9-50858 

37 
38 

656  9.97615 

5 
4 
4 
5 
4 
4 
5 
4 

075  9.53244 

41 
42 
41 
41 
41 
42 

41 

0.46756   347 

II 

32282  9.50896 

94646  9.97610 

34108  9.53285 

0.46715  2.9319 

10 

51 

309  9-50933 

637  9.97606 

140  9.53327 

0.46673   291 

9 

52 

337  9-50970 

37 

% 

37 

627  9,97602 

173  9-53368 

0.46632   263 

8 

53 

364  9.51007 

618  9.97597 

205  9.53409 

0.46591   23s 

7 

54 
55 

392  9.51043 

609  9.97593 

238  9.53450 

0.46550   208 

6 
5 

32419  9.51080 

94599  9.97589 

34270  9.53492 

0.46508  2.9180 

5b 

447  9.51117 

37 

590  9.97584 

303  9.53533 

0.46467   152 

4 

57 

474  9-5"54 

37 

580  9,97580 

41 
41 

0.46426   125 

3 

5« 

502  9.51191 

'i 

571  9.97576 

4 

5 

368  9.53615 

0.46385   097 

2 

^0 

529  9.51227 

561  9.97571 

400  9.53656 

0.46344   070 

I 

557  9.51264 

37 

552  9.97567 

4 

433  9.53697 

0.46303   042 

0 

Nat.CoSLog.  d. 

Nat.  Sin  Log.  d. 

Nat.  Cot  Log. 

c.d. 

Log.Tan  Nat. 

r 

n 


19' 

Nat.  Sin  Log.  d.  Nat.  Cos  Log.  d 


Nat.TanLog. 


c.d. 


Log.  Cot  Nat. 


32557 
584 
612 

639 
667 


9.51264 
9-51301 
9-51338 
9-51374 


32694 
722 
749 
777 
804 


9-51447 
9.51484 
9.51520 
9-51557 
9-51593 


32832 

859 
887 
914 
942 


9.51629 
9.51666 
9.51702 
9.51738 
9-51774 


32969 
997 

33024 
051 
079 


9.51811 
9.51847 
9-51883 
9.51919 
9-51955 


33106 

134 
161 
189 
216 


9.51991 
9.52027 
9.52063 
9.52099 
9-52135 


25 

26 

27 

28 

30 

31 
32 
33 
34 


33244 
271 
298 
326 

__J53_ 

33381 
408 
436 
463 
490 


9.52171 
9.52207 
9.52242 
9.52278 
9-52314 


9-52350 
9-52385 
9-52421 
952456 
9-52492 


33518 
545 
573 
600 
627 


9-52527 
9-52563 
9-52598 
9-52634 
9-52669 


33655 
682 
710 

737 
764 


9-52705 
9.52740 

9-52775 
9.5281 1 
9.52846 


33792 
819 
846 
874 
901 


9.52881 
9.52916 

9-52951 
9.52986 
9.53021 


33929 
956 
983 

3401 1 
038 


953056 
9-53092 
9.53126 
9-53161 
9-53196 


34065 
093 
120 

147 
175 
202 


9-53231 
9.53266 
9-53301 
9-53336 
9-5337? 
9-53405 


94552 
542 
533 
523 
514 


9-97567 
9-97563 
9.97558 
9.97554 
9.97550 


94504 
495 
485 
476 
466 


9-97545 
9-97541 
9.97536 
9.97532 
9.97528 


94457 
447 
438 
428 
418 


9.97523 
9.97519 
9-97515 
9.97510 
9.97506 


94409 
399 
390 
380 
370 


9.97501 
9.97497 
9.97492 
9.97488 
9.97484 


94361 
351 
342 
332 
322 


9-97479 
9-97475 
9.97470 
9.97466 
9.97461 


94313 
303 
293 
284 
274 


9.97457 
9.97453 
9.97448 

9.97444 
9-97439 


94264 
254 
245 
235 
225 


9-97435 
9-97430 
9-97426 
9.97421 
9.97417 


94215 
206 
196 
186 
176 


9.97412 
9.97408 
9-97403 
9-97399 
9-97394 


94167 
157 
147 
137 
127 


9-97390 
997385 
9.97381 
9.97376 
9-97372 


941 18 
108 
098 
088 
078 


9-97367 
9.97363 
9.97358 
9.973.53 
9.97349 


94068 
058 
049 
039 
029 


9.97344 
9.97340 
9.97335 
9.97331 
9.97326 


94019 
009 

93999 
989 
979 
969 


9.97322 

9.97317 
9.97312 
9.97308 
9-97303 
9-97299 


34433 
465 
498 
530 
563 


9.53697 
9.53738 
9.53779 
9.53820 
9.53861 


34596 
628 
661 

693 
726 


9.53902 
9-53943 
9.53984 
954025 
9.54065 


34758 
791 
824 
8^6 


9.54106 

9-54147 
9.54187 
9.54228 
9.54269 


34922 
954 
987 

35020 
052 


9.54309 
9.54350 
9-54390 
9-54431 
9.54471 


35085 
118 
150 
183 
216 


9.54512 
9.54552 
9.54593 
9.54633 
9.54673 


35248 
281 
314 
346 
379 


9.54714 
9.54754 
9.54794 
9.54835 
9-54875 


35412 
445 
477 
510 
543 


9-5491$ 
9-54955 
9.5499$ 
9.5503$ 
9.55075 


35576 
608 
641 

674 
707 


955"$ 
9-5515$ 
9-5519$ 
9.5523$ 
9.55275 


35740 
772 
805 
838 
871 


9-55315 
9-55355 
9-55395 
9-55434 
9.55474 


35904 
937 
969 

36002 
035 


9.55514 
9.55554 
9.55593 
955633 
9.55673 


36068 

lOI 

134 
167 
199 


9-55712 
9.55752 
9.55791 
9.55831 
9.55870 


36232 
265 
298 
331 
364 
397 


9.55910 
9.55949 
9.55989 
9.56028 
9.56067 
9.56107 


0.46303 
046262 
0.46221 
0.46180 
046139 


2.9042 

015 

2.8987 

960 

933 


046098 
046057 
0.46016 
045975 
045935 


2.8905 
878 

851 
824 

797 


045894 

045853 
0.45813 
0.45772 
0-45731 


2.8770 

743 
716 
689 
662 


0.45691 
0.45650 
0.45610 
045569 
0.45529 


2.8636 
609 
582 
556 
529 


0.45488 
0.45448 
0.45407 
o.4$367 
045327 


2.8502 
476 
449 
423 
397 


0.45286 
0.45246 
045206 
045165 
0.45125 


2.8370 
344 
318 
291 
265 


045085 

045045 
045005 
0.44965 
044925 


2.8239 
213 
187 
161 
135 


044885 
0.44845 
0.44805 
0.44765 
0.44725 


2.8109 
083 

057 
032 
006 


0.44685 

044645 
0.44605 
0.44566 
044526 


2.7980 
955 
929 
903 
878 


0.44486 
044446 
0.44407 
0.44367 
044327 


2.7852 
827 
801 
776 
751 


0.44288 
0.44248 
0.44209 
044169 
0.44130 


2.7725 
700 

675 
650 
625 


0.44090 
044051 
04401 1 
043972 
043933 
0.43893 


2.7600 

575 
550 
525 
500 
475 


Nat.  Cos  Log.  d. 


Nat.  Sin  Log.  d. 

70^ 


Nat.  Cot  Log.  c.d.  Log. Tan  Nat. 


Nat.  Sin  Log.  d. 


20^ 

Nat.  Cos  Log.  d. 


Nat. 


;.TanLog.  c.d.  Log.  Cot  Nat 


34202 
229 

257 

284 

3" 


9-53405 
9-53440 
9-53475 
953509 
9-53544 


34339 
366 

393 
421 
448 


9-53578 
9-53613 
9-53647 
9.53682 
9-53716 


34475 
503 
530 

557 
584 


9-53751 
9-53785 
9.53819 
953854 
9-53888 


34612- 

639 
666 
694 
721 


9-53922 
9-53957 
9-53991 
9-54025 

9-54059 


34748 
775 
803 
830 
857 


9-54093 
9.54127 
9.54161 
9-54195 
9-54229 


34884 
912 

939 
966 

993 


9-54263 
9-54297 
9-54335 
9-54365 
9-54399 


35021 
048 

075 
102 
130 


9-54433 
9-54466 
9-54500 
9-54534 
9-54567 


35157 
184 
211 

239 
266 


9-54601 

9-54635 
9-54668 
9-54702 
9-54735 


35293 
320 

347 
375 
402 


9-54769 
9.54802 

9-54836 
9-54869 
9-54903 


35429 
456 
484 
511 
538 


9-54936 
9-54969 
955003 
9-55036 
9-55069 


35565 
592 
619 
647 
674 


9-55102 
9-55136 
9-55169 
9-55202 

9-55235 


36701 
728 

755 
782 
810 
837 


9.55268 
9-55301 
9-55334 
9-55367 
9-55400 
9-55433 


93969 
959 
949 
939 
929 


9-97299 
9-97294 
9-97289 
9-97285 
9.97280 


93919 
909 
899 
889 
879 


9.97276 
9.97271 
9.97266 
9.97262 
9-97257 


93869 
859 
849 
839 
829 


9.97252 
9-97248 
9-97243 
9.97238 

9-97234 


93819 
809 

799 
789 

779 


9-97229 
9-97224 
9.97220 

9-97215 
9.97210 


93769 
759 
748 
738 
728 


9.97206 
9.97201 
9.97196 
9.97192 
9.97187 


93718 
708 
698 
688 
677 


9.97182 
9.97178 

9-97173 
9.97168 

9-97163 


93667 

657 
647 

637 
626 


9-97159 
9-97154 
9.97149 

9-97145 
9.97140 


93616 
606 
596 
585 
575 


9-97135 
9.97130 
9.97126 
9.97121 
9.971 16 


93565 
555 
544 
534 
524 


9.97111 
9.97107 
9.97102 
9-97097 
9-97092 


93514 
503 
493 
483 
472 


9-97087 
9-97083 
9.97078 

9-97073 
9.97068 


93462 

452 
441 

431 
420 


9-97063 
9-97059 
9-97054 
9.97049 

997044 


93410 
400 
389 
379 
368 
358 


9-97039 
9-9703S 
9.97030 
9.9702g 
9.97020 
9.97015 


36397 
430 
463 
496 
529 


9.56107 
9.56146 
9.56185 
9-56224 
9-56264 


36562 

595 
628 
661 
694 


9-56303 
9-56342 
9-56381 
9.56420 
9-56459 


36727 
760 

793 
826 

859 


9-56498 
9-56537 
9-56576 
9-56615 
9-56654 


36892 
925 
958 
991 

37024 


9-56693 
9-56732 
9.56771 
9.56810 
9-56849 


37057 
090 
123 

157 
190 


9-56887 
9.56926 
9-56965 
9-57004 
9-57042 


37223 
256 
289 
322 
355 


9.57081 
9.57120 
9.57158 
9.57197 
9-57235 


37388 
422 

455 
488 
521 


9-57274 
9-57312 
9-57351 
9-57389 
9.57428 


37554 
588 
621 

654 
687 


9-57466 
9-57504 
9-57543 
9-57581 
9-57619 


37720 
754 
787 
820 

853 


9-57658 
9-57696 
9-57734 
9-57772 
9.57810 


37887 
920 

953 

986 

38020 


9-57849 
9.57887 

9-57925 
957963 
9.58001 


38053 
086 
120 

153 
186 


9-58039 
9-58077 
9-58115 
9.58153 
9-58191 


38220 

253 
286 
320 
353 
386 


9-58229 
9.58267 
9.58304 
9-58342 
9.58380 
9.58418 


Nat.  Cos  Log.  d.  Nat.  Sin  Log,  d.  |Nat.  Cot  Log 


0.43893 
0.43854 
0.43815 
0.43776 
0.43736 


2.7475 
450 
425 
400 
376 


0.43697 
0.43658 
0.43619 
0.43580 
0.43541 


2.7351 
326 
302 
277 
253 


0.43502 
0.43463 
0.43424 

0.43385 
043346 


2.7228 
204 
179 

155 
130 


0.43307 
0.43268 
0.43229 
0.43190 
0.43151 


2.7106 
082 
058 

034 
009 


0.43113 
0.4307-^ 
0.43035 
0.42996 
0.42958 


2.6985 
961 
937 
913 


0.42919 
042880 
0.42842 
0.42803 
0.42765 


2.6865 
841 
818 
794 
770 


0.42726 
0.42688 
0.42649 
0.4261 1 
0.42572 


2.6746 
723 
699 
675 
652 


0.42534 
0.42496 

042457 
042419 
0.42381 


2.6628 
605 

581 
558 
534 


0.42342 
0.42304 
0.42266 
0.42228 
0.42190 


2.651 1 
488 
464 
441 
418 


0.42151 
0.42113 
0.42075 
0.42037 
041999 


2.6395 
371 
348 
325 
302 


0.41961 
0.41923 
0.41885 
0.41847 
0.41809 


2.6279 
256 

233 
210 
187 


041771 

041733 
0.41696 
041658 
0.41620 
041582 


2.6165 
142 
119 
096 
074 
051 


c.d.|Log.TanNat.  ' 


69 


Nat.  Sin  Log.    d. 


2r 

Nat.  Cos  Log.  d. 


Nat.TanLog. 


c.d, 


Log.  Cot  Nat, 


30 

31 
32 
33 
34 
35 
36 
37 
38 
39 
40 

41 
42 

43 
44 


35837 
864 
891 
918 
945 


9-55433 
9-55466 

9-55499 
9-55532 
955564 


35973 

36000 

027 

054 
081 


36108 

135 
162 
190 
217 


33 
33 
33 
32 

9-55597  00 
9-55630  i  ti 
9-55663  '  ti 
9-55695  00 
9-55728  f 

9.55761  f 
9-55793  L 
9-55826  1  33 

9.55858 !  3^ 

9-55891  i  ^g 


36244 
271 
298 
325 
352 


36379 
406 

434 
461 


36515 
542 
569 
596 
623 


9-55923  00 

9-55956  a 

9-55988  32 

9.56021  ^^ 

9-56053  3 

9-56085  3 

9-56118  33 

9-56150  3 

9.56182  32 

9.56215  2 


36650 
677 
704 
731 
758 


956247  I2 
9-56279  ^2 
9.563"  ^ 
956343 
9-56375 


36785 
812 

839 
867 
894 


36921 
948 

975 

37002 

029 


32 
32 
32 
33 
32 
32 
32 
32 

9-56568  II 
9-56599  3 
9.56631  i  3 
9.56663  3 
956695  i  II 


9.56408 
9.56440 
9.56472 

9.56504 
9.56536 


37056 
083 
no 

137 
164 


9.56727  L 
9.56759  i  3^ 
9.56790 1 3 
9.56822 1 II 

9.56854 !  II 


37191 
218 

245 
272 

299 


9.56886  3 

9.56917  3J 

9.56949  3 

9.56980  II 
9.57012 


37326 

353 
380 
407 

434 
461 


9.57044 
9.57075 
9.57107 
9-57138 
9-57169 


9-57201 
9-57232 

9-57264  _j 

9-57295  3; 

9-57326  3^ 

9.57358  3^ 


93358 
348 
337 
327 
316 


9.97015 
9.97010 
9.97005 
9.97001 
9.96996 


93306  9-96991 1 

295 

9.96986 

285 

9.96981 

274 

9.96976 

264  9.96971 1 

93253 

9.96966 

243 

9.96962 

232 

996957 

222 

9.96952 

211 

9-96947 

93201 

9.96942 

190 

996937 

180 

9.96932 

169  9.96927 ! 

159 

9.96922 

93148  9.96917 

137 

9.96912 

127 

9.96907 

116 

9-96903 

106 

9.96898 

93095 

l'& 

084 

074 

9.96883 

063  9.96878 

052 

9.96873 

93042 

9.96868 

031 

9-96863 

020 

9-96858 

010 

9-96853 

92999 

9.96848 

92988  9.96843 

978 

9.96838 

967  9.96833 

956 

9.96828 

945 

9.96823 

92935 

9.96818 

924 

9-96813 

913 

9.96808 

902 

9-96803 

892  9.96798 

92881  9.96793 

870  9.96788 

859  9.96783 

849  9.96778 

838  9-96772 

92827  9-96767 

816 

9.96762 

805  9-96757 

794 

9.96752 

784  9-96747 

92773 

762 

751 
740 

729 

718 


9.96742 

9.96737 
9.96732 
9.96727 
9.96722 
9.96717 


38386 

420 
453 
487 
520 


9.58418 
9.58455 
9.58493 
9.58531 
9.58569 


38553  9.58606 

587  9.58644 

620 

9.58681 

654 

9-58719 

687  9.58757 

38721  9.58794 

754 

9.58832 

787  9.58869 

821 

9-58907 

854 

9-58944 

38888  9.58981 

921 

9.59019 

955 

9.59056 

988 

9-59094 

39022  9.59131 

39055 

9-59168 

089  9.59205 

122 

9-59243 

156  9.59280 

igo 

9-59317 

39223 

9-59354 

257 

9-59391 

290  9.59429 

324 

9-59466 

357 

9-59503 

39391 

9-59540 

425 

9-59577 

458 

9.59614 

492 

9-59651 

526  9.5088 

39559 

9-59725 

593 

9.59762 

626 

9-59799 

660 

9.59835 

694  9-59872 

39727 

9.59909 

761  9.59946 

795 

9.59983 

829  9.60019 

862 

9.60056 

39896 

9.60093 

930 

9.60130 

963 

9.60166 

997 

9.60203 

40031 

9.60240 

40065 

9.60276 

098 

9.60313 

132 

9-60349 

166 

9.60386 

200 

9.60422 

40234 

267 
301 
335 
369 
403 


9-60459 
9.60495 
9-60532 
9.60568 
9.60605 
9.60641 


0.41582 

0.41545 
0.41507 
0.41469 
0.41431 


2.6051 
028 
006 

2.5983 
961 


0.41394 
0.41356 
0.41319 
0.41281 
0-41243 


2.5938 
916 
893 
871 


0.41206 
0.41 168 
0.41131 
0.41093 
0.41056 


2.5826 
804 
782 
759 
737 


0.41019 
0.40981 
0.40944 
0.40906 
0.40869 


2.5715 
693 
671 
649 
627 


0.40832 

0-40795 
0.40757 
0.40720 
0.40683 


2.5605 
583 
561 
539 
517 


0.40646 
0.40609 
040571 
040534 
040497 


2.5495 
473 
452 
430 
408 


040460 
0.40423 
0.40386 
0.40349 
040312 


2.5386 
365 
343 
322 
300 


0.40275 
040238 
040201 
0.40165 
0.40128 


2.5279 

257 
236 
214 
193 


0.40091 
0.40054 
0.40017 
0.39981 
0.39944 


2.5172 
150 
129 
108 
086 


0.39907 
0.39870 
0.39834 
0.39797 
0.39760 


2.5065 
044 
023 
002 

2.4981 


0.39724 
0.39687 
0.39651 
0.39614 
0.39578 


2.4960 

939 
918 

897 
876 


0.39541 
O.3950S 
0.39468 
0.39432 
0.39395 
0.39359 


24855 
834 
813 
792 
772 
751 


Nat.  Cot  Log.  c.d.  Log.Tan  Nat 


Nat.CoSLog.   d. 


Nat.  Sin  Log.   d. 


68^ 


22° 

'   Nat.  Sin  Log.  d.  Nat.  Cos  Log.  d.  Nat.TanLog. 


c.d.  Log.  Cot  Nat 


37461 
488 
515 

542 
569 


9-57358 
9-57389 
9.57420 

9-57451 
9-57482 


37595 
622 
649 
676 
703 


9-57514 
9.57545 
9-57576 
9-57607 
9-57638 


37730 
757 
784 
811 
838 


9-57669 
9.57700 

9-57731 
9.57762 

9-57793 


37865 
892 
919 
946 
973 


9.57824 

9-57855 
9-57885 
9.57916 
9-57947 


37999 
38026 

053 
080 
107 


9-57978 
9.58008 

9.58039 
9-58070 
9.58101 


38134 
161 

188 

215 
241 


9-58131 
9.58162 
9.58192 
9.58223 
9-58253 


38268 

295 
322 

349 
376 


9-58284 
9-58314 
9-58345 
958375 
9-58406 


38403 
430 
456 
483 
510 


9.58436 
9.58467 
9.58497 
9.58527 
9.58557 


38537 
564 
591 
617 

644 


9.58588 
9.58618 
9.58648 
9.58678 
9.58709 


38671 
698 
725 
752 
778 


9.58739 
9.58769 
9-58799 
9.58829 

9-58859 


38805 
832 
859 
886 
912 


9.58889 
9.58919 
9.58949 
9.58979 
9.59009 


38939 
966 

993 

39020 

046 

073 


9.59039 
9.59069 
9.59098 
9.59128 
9-59158 
9.59188 


92718 
707 
697 
686 
675 


9.96717 
9.96711 
9.96706 
9.96701 
9.96696 


92664 

653 
642 

631 
620 


9.96691 
9.96686 
9.96681 
9.96676 
9.96670 


92609 

598 
587 
576 
565 


9.96665 
9.96660 
9.96655 
9.96650 
9.96645 


92554 
543 
532 
521 
510 


9.96640 

9.96634 
9.96629 
9.96624 
9.96619 


92499 
488 

477 
466 

455 


9.96614 
9.96608 
9.96603 
9.96598 
996593 


92444 
432 
421 
410 
399 


9.96588 
9.96582 

9.96577 
9.96572 

9.96567 


92388 

377 
366 

355 
343 


9.96562 
9.96556 
9.96551 
9.96546 
9.96541 


92332 
321 
310 
299 
287 


9-96535 
9.96530 

9-96525 
9.96520 
9.96514 


92276 
265 
254 
243 
231 


9-96509 
9-96504 
9.96498 

9-96493 
9.96488 


92220 
209 
198 
186 
175 


9.96483 
9.96477 
9.96472 
9.96467 
9.96461 


92164 
152 
141 
130 
119 


9.96456 
9.96451 

9.96445 
9.96440 

9.96435 


92107 
096 
085 

073 
062 
050 


9.96429 
9.96424 
9.96419 
9.96413 
9.96408 
9.96403 


40403 
436 
470 
504 
538 


9.60641 
9.60677 
9.60714 
9.60750 
9.60786 


40572 

9.60823 

606 

9.60859 

640  9.60895 

674  9.60931 

707 

9.60967 

40741 

9.61004 

775 

9.61040 

809  9.61076 

843 

9.61112 

877  9.61148 

409H 

9.61184 

945 

9.61220 

979 

9.61256 

41013 

9.61292 

047 

9.61328 

4108 1 

9.61364 

"5 

9.61400 

149 

9.61436 

183  9.61472 

217 

9.61508 

41251 

9.61544 

285  9.61579 

319 

9.61615 

9.61651 
9.61687 

41421 

9.61722 

455 

9.61758 

490 

9.61794 

524 

9.61830 

558  9.61865 

41592 

9.61901 

9.61936 

660 

9.61972 

694 

9.62008 

728  9.62043 

41763  9.62079 

797 

9.62114 

831 

9.62150 

865  9.62185 

899 

9.62221 

41933 

9.62256 

968  9.62292 

42002 

9-62327 

036  9.62362 

070 

9-62398 

42105 

9-62433 

139 

9.62468 

173 

9.62504 

207 

9-62539 

242 

9-62574 

42276 
310 
345 
379 
413 
447 


9.62609 
9-62645 
9.62680 
9.62715 
9.62750 
9.62785 


0.39359 
0.39323 
0.39286 
0.39250 
0.39214 


2.4751 
730 
709 
689 
668 


0.39177 
0.39141 
0.39105 
0.39069 
0.39033 


2.4648 
627 
606 

586 
566 


0.38996 
0.38960 

0.38924 
0.38888 
0.38852 


2.4545 
525 
504 
484 

464 


0.38816 
0.38780 

0.38744 
0.38708 
0.38672 


2.4443 
423 
403 
383 
362 


0.38636 
0.38600 
0.38564 
0.38528 
0.38492 


2.4342 
322 
302 
282 
262 


0.38456 
0.38421 
0.3838.5 
0.38349 
0.38313 


2.4242 
222 
202 
182 
162 


0.38278 
0.38242 
0.38206 
0.38170 
0.38135 


2.4142 
122 
102 
083 
063 


0.38099 
0.38064 
0.38028 
0.37992 
0.37957 


2.4043 
023 
004 

2.3984 
964 


0.37921 
0.37886 
0.37850 
0.37815 
0.37779 


2.3945 
925 
906 
886 
867 


0.37744 
0.37708 
0.37673 
0.37638 
0.37602 


2.3847 
828 
808 
789 
770 


0.37567 
0.37532 
0.37496 
0.37461 
0.37426 


2.3750 
731 
712 

693 
673 


0.37391 
0.37355 
0.37320 

0.37285 
0.37250 
0.37215 


2.3654 
635 
616 

597 
578 
559 


Nat.  Cot  Log.  c.d.  Log. Tan  Nat, 


Nat.  Cos  Log.  d.  Nat.  Sin  Log.  d 


67° 


23° 


Nat.  Sin  Log.  d. 


Nat.  Cos  Log.  d. 


Nat  Tan  Log.  c.d.Log.  Cot  Nat, 


39073 
100 
127 

153 
180 


9.59188 
9.59218 
959247 
959277 
9-59307 


39207 

234 
260 
287 
314 


9-59336 
9-5936<3 
959396 
959425 
9-59455 


39341 
367 
394 
421 
448 


9-59484 
9-59514 
9-59543 
9-59573 
9-59602 


39474 
501 

528 

555 
581 


9-59632 
9-5061 
9.59690 
9.59720 
9-59749 


39608 

635 
661 
688 
715 


9-59778 
9.59808 

9-59837 
9-59866 

9-59895 


39741 
768 

795 
822 
848 


9-59924 
9-59954 
9-59983 
9,60012 
9.60041 


39875 
902 
928 

955 
082 


9.60070 
9.60099 
9.60128 
9.60157 
9.60186 


40008 

035 
062 
088 
115 


9.60215 
9.60244 
9.60273 
9.60302 
9.60331 


40141 

168 

195 
221 
248 


9-60359 
9.60388 
9.60417 
9.60446 
9-60474 


40275 
301 
328 
355 
381 


9-60503 
9-60532 
9.60561 
9-60589 
9.60618 


40408 

434 
461 
488 
514 


9.60646 
9-60675 
9.60704 
9.60732 
9.60761 


40541 
567 
594 
621 
647 
674 


9.60789 
9.60818 
9.60846 
9.60875 
9-60903 
9-60931 


92050 
039 
028 
016 
005 


9.96403 
9-96397 
9-96392 
9-96387 
9.96381 


91994 

9.96376 

982 

9-96370 

971 

9-96365 

959 

9-96360 

948  9.96354 

91936  9-96349 

925 

9.96343 

914 

9-96338 

902 

996333 

891 

9-96327 

91879  9.96322 

868 

9.96316 

856  9-9631 1 

845 

9.96305 

833 

9-96300 

91822 

9-96294 

810 

9.96289 

799 

9.96284 

787  9.96278 

775 

9-96273 

91764  9.96267 

752 

9.96262 

741 

9.96256 

729 

9.96251 

718  9.96245 

91706  9.96240 

694  9.96234 

683  9.96229 

671 

9.96223 

660 

9.96218 

91648 

9.96212 

636  9.96207 

625  9.96201 

613  9.96196 

601 

9.96190 

91590 

9-96185 

578  9-96179 

566  9-96174 

555 

9.96168 

543 

9.96162 

91531 

9.96157 

519 

9.96151 

508  9.96146 

496  9.96140 

484 

996135 

91472 

9.96129 

461 

9.96123 

449 

9.96118 

437 

9.961 12 

425 

9.96107 

9I4I4 

402 

390 
378 
366 

355 


9.96101 

9-96095 
9.96090 
9.96084 
9.96079 
9-96073 


Nat.  Cos  Log.  d. 


Nat.  Sin 


42447 
482 

516 

551 
585 


9-62785 
9.62820 

9.62855 

9.62890 
9.62926 


42619 

654 

688 
722 
757 


9.62961 
9.62996 
9.63031 
9.63066 
9.63101 


42791 
826 
860 
894 
929 


9-63135 
9.63170 
9.63205 
9.63240 
9-63275 


42963 
998 

43032 
067 
loi 


9.63310 
9-63345 
9-63379 
9.63414 
9.63449 


43136 
170 
205 

239 
274 


9.63484 
9-63519 
9.63553 
9.63588 
9.63623 


43308 
343 
378 
412 
447 


9.63657 
9.63692 
9.63726 
9.63761 
9.63796 


43481 
516 
550 
585 
620 


9.63830 
9-63865 
9-63899 
9-63934 
9.63968 


43654 
689 
724 
758 
793 


9-64003 
9.64037 
9.64072 
9.64106 
9.64140 


43828 
862 
897 
932 
966 


9.64175 
9.64209 

9-64243 
9.64278 
9.64312 


44001 
036 
071 
105 
140 


9-64346 
9.64381 
9.64415 

9-64449 
9.64483 


44175 
210 
244 
279 
314 


9.64517 
9-64552 
9-64586 
9.64620 
9.64654 


44349 
384 
418 

453 
488 

523 


9.64688 
9.64722 
9.64756 
9.64790 
9.64824 
9.64858 


)073  I 523  9.04050  ^'    0.35142   400 

Log.  d.  Nat.CotLog.  c.d.|Log.TanNat 


0-37215 
0.37180 

0.37145 
0.371 10 

0.37074 


2.3559 
539 
520 
501 
483 


0.37039 
0.37004 
0.36969 

0.36934 
0.36899 


2.3464 

445 
426 
407 


0.36865 
0.36830 

0.36795 
0.36760 
0.36725 


2.3369 
351 
332 
313 

294 


0.36690 
0.36655 
0.36621 
0.36586 
0.36551 


2.3276 
257 
238 
220 
201 


0.36516 
0.36481 
0.36447 
0.36412 
0.36377 


2.3183 
164 
146 
127 
109 


0.36343 
0.36308 
0.36274 
0.36239 
0.36204 


2.3090 
072 
053 
035 
017 


0.36170 

0.36135 
0.36101 
0.36066 
0.36032 


2.2998 
980 
962 
944 
925 


0.35997  2.2907 

0.35963 

0.35928 

0.35894 

0.35860 


871 
853 
835 


0.35825 
0.35791 
0-35757 
0.35722 
0.35688 


2.2817 

799 
781 

763 
745 


0-35654 
0.35619 

0.35585 
0.35551 
0.35517 


2.2727 

709 
691 

673 
655 


0.35483 
0.35448 
0.35414 
0.35380 
0.35346 


2.2637 
620 
602 

584 
566 


0.35312 
0.35278 
0.35244 
0.35210 
0.35176 
0.35142 


2.2549 
531 
513 
496 
478 
460 


24 

0 

f 

Nat.  Sin  Log.  d. 

Nat.  Cos  Log.  d. 

Nat.TanLog. 

c.d.  Log.  Cot  Nat. 

0 

40674  9.60931 

29 

91355  9-96073 

f. 

44523  9.64858 

34 
34 
34 
34 
34 
34 

0.35142  2.2460 

60 

I 

700  9.60960 

343  9-96067 

5 

558  9.64892 

0.35108   443 

59 

2 

727  9.60988 

o9 

331  9-96062 

593  9-64926 

0.35074   425 

58 

3 

753  9.61016 

29 
28 

319  9-96056 

A 

627  9.64960 

0.35040   408 

57 

4 
6 

780  9.61045 

307  9.96050 

5 
6 

662  9.64994 

0.35006   390 

56 
55 

40806  9.61073 

91295  9.96045 

44697  9.65028 

0.34972  2.2373 

6 

833  9.61  lOI 

08 

283  9.96039 

732  9.65062 

0-34938   355 

54 

7 

860  9.61 129 

29 
08 

272  9.96034 

767  9.65096 

34 

0.34904   338 

53 

8 

886  9.61158 

260  9.96028 

f. 

802  9.65130 

34 

0.34870   320 

52 

9 

913  9.61186 

28 

248  9.96022 

5 
6 

837  9.65164 

34 
33 
34 
34 
34 

0.34836   303 

51 

10 

40939  9.61214 

91236  9.96017 

44872  9.65197 

0.34803  2.2286 

50 

II 

966  9.61242 

oO 

224  9.9601 1 

5 

907  9.65231 

0.34769   268 

49 

12 

992  9.61270 

03 

212  9.96005 

942  9.65265 

0.34735   251 

48 

13 

41019  9.61298 

og 

200  9.96000 

977  9.65299 

0.34701   234 

47 

14 

04s  9.61326 

28 

08 

188  9.95994 

I 

45012  9-65333 

34 
33 

0.34667   216 

46 
'45 

15 

41072  9.61354 

91176  9.95988 

45047  9-65366 

0.34634  2.2199 

lb 

098  9.61382 

164  9.95982 

082  9.65400 

0.34600   182 

44 

17 

125  9.61411 

27 

08 

152  9-95977 

117  9-65434 

34 
33 

11 

33 
34 
34 
33 
34 
33 
34 
33 

0.34566   165 

43 

l8 

151  9.61438 

140  9-95971 

6 

152  9.65467 

0.34533   148 

42 

19 

178  9.61466 

28 

28 

128  9.95965 

5 
6 

187  9.65501 

0.34499   130 

41 
40 

20 

41204  9.61494 

91 116  9.95960 

45222  9.65535 

0.34465  2.2113 

21 

231  9.61522 

28 

104  9.95954 

6 

257  9-65568 

0.34432   096 

39 

22 

257  9.61550 

28 

092  9.95948 

5 

292  9.65602 

0.34398   079 

38 

23 

284  9.61578 

28 

080  9.95942 

5 
6 
6 

,327  9.65636 

0.34364   062 

37 

24 

310  9.61606 

28 

og 

068  9-95937 

362  9.65669 

0-34331   045 

36 
35 

25 

41337  9.61634 

91056  9.95931 

45397  9.65703 

0.34297  2.2028 

26 

363  9.61662 

27 

og 

044  9-95925 

5 
6 

432  9.65736 

0.34264   on 

34 

27 

390  9.61689 

032  9.95920 

467  9.65770 

0.34230  2.1994 

33 

28 

416  9.61717 

28 
28 

020  9.95914 

f. 

502  9.65803 

0.34197   977 

32 

29 

443  9-61745 

008  9.95908 

6 

5 
5 

538  9.65837 

34 
33 

0.34163   960 

.31 

30 

41469  9.61773 

90996  9.95902 

45573  9.65870 

0.34130  2.1943 

30 

31 

496  9.61800 

oQ 

984  9.95897 

608  9.65904 

33 
34 

0.34096   926 

29 

32 

522  9.61828 

08 

972  9-95891 

(=, 

643  9.65937 

0.34063   909 

28 

3S 

549  9-61856 

960  9.95885 

5 

678  9.65971 

0.34029   892 

27 

34 

575  9-61883 

27 
28 

og 

948  9.95879 

6 

5 

713  9.66004 

33 
34 
33 

0-33996   876 

2b 

35 

41602  9.61911 

90936  9.95873 

45748  9.66038 

0.33962  2.1859 

25 

36 

628 .  9.61939 

924  9.95868 

784  9.66071 

0-33929   842 

24 

37 

655  9.61966 

^8 

911  9.95862 

6 

819  9.66104 

33 

0.33896   825 

23 

3B 

681  9.61994 

899  9.95856 

6 

854  9.66138 

34 
33 
33 
34 

0.33862   808 

22 

39 

707  9.62021 

27 
28 

887  9.95850 

6 

5 

A 

889  9.66171 

0.33829   792 

21 
20 

40 

41734  9.62049 

90875  9-95844 

45924  9.66204 

0.33796  2.1775 

41 

760  9.62076 

08 

863  9.95839 

960  9.66238 

0.33762   758 

19 

42 

787  9.62104 

851  9-95833 

A 

995  9.66271 

0.33729   742 

18 

43 

813  9.62131 

08 

839  9.95827 

5 

46030  9.66304 

33 

34 
33 

0.33696   725 

17 

44 
45 

840  9.62159 

27 

28 

826  9.95821 

6 

065  9.66337 

0.33663   708 

lb 

41866  9.62186 

90814  9.95815 

46101  9.66371 

0.33629  2.1692 

15 

46 

892  9.62214 

802  9.95810 

136  9.66404 

0.33596   675 

14 

47 

919  9.62241 

27 

790  9.95804 

6 

171  9.66437 

33 
33 
33 
34 
33 

0.33563   659 

13 

48 

945  9.62268 

27 
28 
27 

778  9-95798 

6 

206  9.66470 

0.33530   642 

12 

49 

972  9.62296 

766  9.95792 

6 

242  9.66503 

0.33497   625 

II 

lo 

50 

41998  9.62323 

90753  9.95786 

46277  9.66537 

0.33463  2.1609 

=^1 

42024  9.62350 

27 

741  9.95780 

5 

312  9.66570 

0.33430   59? 

9 

52 

051  9-62377 

28 

729  9.95775 

348  9.66603 

33 
33 
33 
33 

0.33397   576 

8 

,S3 

077  9.62405 

717  9.95769 

A 

383  9.66636 

0.33364   560 

7 

54 
55 

104  9.62432 

27 

704  9.95763 

6 
6 

418  9.66669 

0.33331   543 

b 
~5 

42130  9-62459 

90692  9.95757 

46454  9.66702 

0.33298  2.1527 

,0 

156  9.62486 

27 

680  9.95751 

5 

489  9.66735 

0.3.3265   510 

4 

57 

183  9-62513 

27 
28 

668  9.95745 

6 

525  966768 

33 
33 
33 
33 

0.33232   494 

3 

5« 

209  9.62541 

655  9.95739 

6 

560  9.66801 

0.33199   478 

2 

^0 

235  9-62568 

27 

643  9-95733 
631  9-95728 

5 

595  9-66834 

0.33166   461 

I 

262  9.62595 

27 

631  9.66867 

0.33133   445 

0 

Nat.CoSLog.  d. 

Nat.  Sin  Log.  d. 

Nat.  Cot  Log. 

c.d. 

Log.Tan  Nat. 

r 

66' 


26° 


f   Nat.  Sin  Log.  d.  Nat.  Cos  Log.  d.  Nat.TanLog.  c.d.  Log.CotNat. 


42262 
288 
315 
341 
367 


9-62595 
9.62622 
9.62649 
9.62676 
9-62703 


42394 
420 
446 
473 
499 


9.62730 
9.62757 
9.62784 
9.6281 1 
9.62838 


42525 
552 
578 
604 
631 


42657 
683 
709 
736 
762 


42788 

815 
841 
867 
894 


42920 
946 
972 
999 

43025 


27 
27 
27 
27 
27 

27 
27 

27 
27 
27 
27 

I  26 
i  27 

;  27 

27 

j27 

26 

1 27 
I  27 

9-63133  26 

9-63159   27 
9.63186  % 

9-63213  :  26 

9-63239  !  27 

26 


9.62865 
9.62892 
9.62918 

9.62945 

9.62972 


9.62999 

9.63026 
9.63052 
9.63079 

9.63106 


43051 
077 

104 

130 

156 


9.63266 

9-63292  i  27 
9-63319  26 
963345   27 


9-63398  ^  27 
9-63425   26 
9-63451 
9.63478 

9-63504 


43182 
209 

235 
261 
287 


9-63531 
9-63557 
9-63583 
9.63610 
963636 


43313 
340 
366 
392 
418 


9.63662 
9.63689 

9-63715 
9.63741 

963767 


43445 
471 
497 
523 
549 


9-63794 
9.63820 
9.63846 
9-63872 
9-63898 


43575 
602 
628 

654 
680 


9.63924 
9-63950 
9-63976 
9.64002 
9.64028 


43706 
733 

759 
785 
811 

837 


9.64054 
9.64080 
9.64106 
9.64132 
9.64158 
9.64184 


90631 
618 
606 

594 
582 


9.95728 
9-95722 
9.95716 
9.95710 
9-95704 


90569  9.95698 

557 

9-95692 

545 

9.95686 

532 

9.95680 

520 

9-95674 

90507 

9-95668 

495 

995663 

483 

995657 

470 

9-95651 

458 

9-95645 

90446  9.95639 

433 

995633 

421 

9-95627 

408 

9.95621 

396  9.95615 

90383  9-95609 

371 

9-95603 

358 

9-95597 

346 

9-95591 

334 

9-95585 

90321 

9-95579 

309 

9-95573 

296 

9-95567 

284  9-95561 

271 

9-95555 

90259  9.95549 

246  9-95543 

233 

9-95537 

221 

9-95531 

208 

9-95525 

90196 

9-95519 

183  9-95513 

171 

9-95507 

158 

9-95500 

14b 

9-95494 

90133 

9.95488 

120 

9.95482 

108 

9-95476 

095 

9-95470 

082 

9.95464 

90070 

9-95458 

057 

9-95452 

045 

9.95446 

032 

9.95440 

019 

9-95434 

90007 

995427 

89994 

9-95421 

981 
■§68 

9-95415 

9.95409 

956  9-95403 

89943 

9^0 
918 
905 

892 

879 


9-95397 
9-95391 
995384 
9.95378 
995372 
995366 


46631 
666 
702 

737 
772 


9.66867 
9.66900 
9.66933 
9.66966 
9.66999 


46808 
843 
879 
914 

950 


9.67032 
9-67065 
9.67098 
9.67131 
9-67163 


46985 

47021 

056 

092 

128 


9.67196 
9.67229 
9.67262 
9.67295 
9-67327 


47163 
199 

234 
270 

305 


9.67360 

9-67393 
9.67426 
9.67458 
9.67491 


47341 
377 
412 

448 
483 


9-67524 
9-67556 
9.67589 
9.67622 
9-67654 


47519 
555 
590 
626 
662 


9-67687 
9.67719 
9.67752 
9.67785 
9.67817 


47698 

733 
769 
805 
840 


9.67850 
9.67882 
9.67915 

9-67947 
9.67980 


47876 
912 
948 
984 

48019 


9.68012 
9.68044 
9.68077 
9.68109 
9.68142 


48055 
091 
127 
163 
iq8 


9.68174 
9.68206 
9-68239 
9.68271 
9-68303 


48234 
270 
306 
342 
378 


9.68336 
9.68368 
9.68400 
9.68432 
968465 


48414 

450 
486 
521 
557 


9.68497 
9.68529 
9.68561 

9-68593 
9.68626 


48593 
629 
665 
701 
737 
773 


9.68658 
9.68690 
9.68722 

9-68754 
9.68786 
9.68818 


0.33133 
0.33100 
0.33067 

0-33034 
0.33001 


2.1445 
429 
413 
396 
380 


0.32968 

0-32935 
0.32902 
0.32869 
0.32837 


2.1364 
348 
332 
315 
299 


0.32804 
0.32771 
0.32738 
0.32705 
0.32673 


2.1283 
267 
251 
235 
219 


0.32640 
0.32607 

0.32574 
0.32542 

0-32509 


2.1203 

187 
171 
IS5 
139 


0.32476 
0-32444 
0.32411 

0.32378 
0.32346 


2.1123 
107 
092 
076 
060 


0.32313 
0.32281 
0.32248 
0.32215 
0.32183 


2.1044 
028 
013 

2.0997 
981 


0.32150 
0.32118 
0.32085 
0.32053 
0.32020 


2.0965 
950 
934 
918 

903 


0.31988 
0.31956 
0.31923 
0.31891 
0.31858 


2.0887 
872 
856 
840 
825 


0.31826 
0.31794 
0.31761 
0.31729 
0.31697 


2.0809 

794 
778 

763 

748 


0.31664 
0.31632 
0.31600 
0.31568 
0.31535 


2.0732 
717 
701 
686 
671 


0.31503 
0.31471 

0.31439 
0.31407 

0.31374 


2.0655 
640 
625 
609 
594 


0.31342 
0.31310 
0.31278 
0.31246 
0.31214 
0.31182 


2.0579 

564 
549 
533 
518 
503 


Nat.  Cot  Log.  c.d.  Log. Tan  Nat. 


60 

59 
58 
57 
5^ 
55 
54 
53 
52 

50 

49 
48 
47 

45 

44 
43 
42 

40 

39 
38 
37 
_36 
35 
34 
33 
32 


Nat.  Cos  Log.  d. 


Nat.  Sin  Log.  d. 

64^ 


< 

26 

0 

f 

Nat.  Sin  Log.  d. 

Nat.  Cos  Log.  d. 

Nat.TanLog. 

c.d.  Log.  Cot  Nat. 

0 

43837  9-64184 

89879  9-95366 

5 

48773  9.68818 

32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
31 
32 

0.31 182  2.0503 

60 

I 

863  9.64210 

05 

867  995360 

5 

809  9.68850 

0.31 150   488 

59 

2 

889  9.64236 

05 

854  9-95354 

5 

0.31118   473 

58 

3 

916  9.64262 

06 

841  9.95348 

881  9.68914 

0.31086   458 

57 

4 

T 

942  9.64288 

25 

05 

828  9.95341 

7 
6 

917  9-68946 

0.31054   443 

5t) 

43968  9.64313 

89816  9.95335 

48953  9-68978 

0.31022  2.0428 

55 

6 

06 

803  9-95329 

f. 

989  9.69010 

0.30990   413 

54 

7 

44020  9.64365 

05 

790  9-95323 

f. 

49026  9.69042 

0.30958   398 

53 

8 

046  9.64391 

-^6 

777   9-95317 

062  9.69074 

0.30926   383 

52 

9 

072  9.64417 

25 

06 

764  9-95310 

6 
5 

098  9.69106 

0.30894   368 

51 

10 

44098  9.64442 

89752  9-95304 

49134  9.69138 

0.30862  2.0353 

50 

II 

124  9.64468 

05 

739  9-95298 

f. 

170  9.69170 

0.30830   338 

49 

12 

151  9-64494 

% 

726  9.95292 

6 

206  9.69202 

0.30798   323 

48 

13 

177  9-64519 

713  9.95286 

242  9.69234 

0.30766   308 

47 

14 

203  9.64545 

26 

% 

700  9-95279 

7 
6 

278  9.69266 

0.30734   293 

4b 
45 

15 

44229  9.64571 

89687  9.95273 

49315  9-69298 

0.30702  2.0278 

I6 

255  9-64596 

674  995267 

A 

351  9.69329 

0.30671   263 

44 

17 

281  9.64622 

% 

662  9.95261 

387  9-69361 

0.30639   248 

43 

i8 

307  9-64647 

649  9-95254 

7 
6 

423  9-69393 

32 
32 
32 
31 

0.30607   233 

42 

19 

333  9-64673 

25 

"6 

636  9.95248 

6 
6 

459  9-69425 

0.30575   219 

4i 

20 

44359  9-64698 

89623  9-95242 

49495  9-69457 

0.30543  2.0204 

40 

21 

385  9-64724 

% 

610  9.95236 

532  9.69488 

0.30512   189 

39 

22 

411  9-64749 

597  9-95229 

I 
6 
6 

7 
5 

568  9.69520 

32 

0.30480   174 

38 

23 

437  9-64775 

25 
26 
25 

584  9.95223 

604  9-69552 

32 

0.30448   160 

37 

24 

464  9.64800 

571  9-95217 

640  9-69584 

32 
31 
32 
32 

0.30416   145 

3t^ 
35 

25 

44490  9.64826 

89558  9-95211 

49677  9.69615 

0.30385  2.0130 

2b 

516  9.64851 

545  9-95204 

713  9.69647 

0.30353   "5 

34 

27 

542  9-64877 

25 
25 
26 

25 
25 

532  9.95198 

6 

749  9-69679 

0.30321    lOI 

33 

28 

568  9.64902 

519  9-95192 

786  9.69710 

31 

0.30290   086 

32 

29 

30 

594  9-64927 

506  9-95185 

7 
6 

6 

822  9.69742 

32 
32 
31 

0.30258   072 

31 
30 

44620  9.64953 

89493  9-95179 

49858  9.6977^ 

0.30226  2.0057 

31 

646  9.64978 

480  9-95173 

5 

894  9.69805 

0.30195   042 

29 

32 

672  9.65003 

467  9-95167 

931  9.69837 

0.30163   028 

28 

33 

698  9.65029 

25 
25 
25 

"6 

454  9-95160 

^ 

967  9.69868 

0.30132   013 

27 

34 
35 

724  9.65054 

441  9-95154 

6 

50004  9.69900 

32 
32 

0.30100  1.9999 

2b 

25 

44750  9-65079 

89428  9.95148 

50040  9.69932 

0.30068  1.9984 

3t> 

776  9.65104 

415  9-95141 

^ 

076  9-69963 

31 
32 

0.30037   970 

24 

37 

802  9.65130 

25 
25 
25 
25 
25 

402  9-95135 

5 

113  9-69995 

0.30005   955 

23 

3a 

828  9.65155 

389  9-95129 

149  9.70026 

31 

0.29974   941 

22 

39 

854  9-65180 

376  9.95122 

7 
6 

185  9.70058 

32 
31 

0.29942   926 

21 

40 

44880  9.65205 

89363  9-95116 

50222  9.70089 

0.29911  1.9912 

20 

41 

906  9.65230 

350  9.951 10 

258  9.70121 

32 

0.29879   897 

19 

42 

932  9-65255 

337  9-95103 

6 

295  9-70152 

31 

32 

0.29848   883 

18 

43 

958  9.65281 

25 

25 
25 

25 

324  995097 

331  9.70184 

0.29816   868 

17 

44 

984  9.65306 

311  9.95090 

6 
6 

I 

368  9.70215 

31 
32 

0.29785   854 

lb 
15 

45 

45010  9.65331 

89298  9.95084 

50404  9.70247 

0.29753  1.9840 

46 

062  9.65381 

285  9-95078 

441  9.70278 

31 
31 

0.29722   825 

14 

47 

272  9-95071 

477  9-70309 

0.29691   811 

13 

48 

088  9.65406 

25 
25 
25 
25 
25 
25 
25 
24 

259  9.95065 

5 

514  9-70341 

32 
31 
32 

0.29659   797 

12 

49 
50 

114  9-65431 

245  9-95059 

7 
6 

550  9-70372 

0.29628   782 

II 

45140  9.65456 

89232  9.95052 

50587  9.70404 

0.29596  1.9768 

10 

51 

166  9.6548X 

219  9.95046 

623  9-70435 

31 

0.29565   754 

9 

52 

192  9-65506 

206  9.95039 

I 
5 

660  9.70466 

31 

0.29534   740 

8 

53 

218  9-65531 

193  9-95033 

696  9-70498 

32 
31 
31 

0.29502   725 

7 

54 

243  9-65556 

180  9.95027 

7 
6 

733  9-70529 

0.29471   7" 

b 

65 

45269  9.65580 

89167  9.95020 

50769  9.70560 

0.29440  1.9697 

6 

56 

25 
25 

153  9-95014 

806  9.70592 

32 
31 

0.29408   683 

4 

57 

321  9-65630 

140  9-95007 

6 
6 

843  970623 

0.29377   669 

3 

5a 

347  9-65655 
373  9-65680 

25 
25 

127  9.95001 

879  9.70654 

31 
31 

0.29346   654 

2 

U 

114  9-94995 

^ 

916  9.70685 

0.29315   640 

I 

399  9-65705 

25 

loi  9-94988 

' 

953  9-70717 

32 

0.29283   626 

0 

Nat.  Cos  Log.  d. 

Nat.  Sin  Log.  d. 

Nat.  Cot  Log. 

c.d. 

Log.Tan  Nat. 

f 

63 

27' 

D 

/ 

Nat.  Sin  Log.  d. 

Nat.  Cos  Log.  d.| 

Nat  Tan  Log. 

c.d. 

Log.  Cot  Nat. 

0 

45399  9-65705 

24 
25 
25 
25 
24 
25 
25 

89101  9.94988 

5 

50953  9.70717 

31 
31 
31 
31 
32 
31 
31 

0.29283  1.9626 

60 

I 

425  9-65729 

087  9.94982 

7 
5 

989  9.70748 

0.29252   612 

59 

2 

451  9.65754 

074  9.94975 

51026  9.70779 

0.29221   598 

58 

3 

477  9-65779 

061  9.94969 

063  9.70810 

0.29190   584 

57 

4 

503  9.65804 

048  9.94962 

7 
6 

099  9.70841 

0.29159   570 

56 

5 

45529  9.65828 

89035  9.94956 

5II36  9-70873 

0.29127  1.9556 

55 

6 

554  9-65853 

021  9.94949 

6 

173  9.70904 

0.29096   542 

54 

7 

580  9.65878 

008  9.94943 

209  9.70935 

0.29065   528 

53 

8 

606  9.65902 

24 
25 
25 
24 
25 

88995  9.94936 

7 

6 

7 

31 
31 
31 
31 
31 
31 

0.29034   514 

52 

9 

632  9.65927 

981  9.94930 

283  9.70997 

0.29003   500 

51 
50 

10 

45658  9-65952 

88968  9.94923 

5I3I9  9.71028 

0.28972  1.9486 

II 

684  9-65976 

955  9-94917 

(\ 

356  9.71059 

0.28941   472 

49 

12 

710  9.66001 

942  9.949" 

393  9-71090 

0.28910   458 

48 

13 

736  9.66025 

928  9-94904 

I 

7 
6 

430  9.71121 

0.28879   444 

47 

14 

762  9.66050 

25 
25 

915  9.94898 

467  9.71153 

32 
31 
31 
31 
31 

0.28847   430 

46 

15 

45787  9.66075 

88902  9.94891 

S1503  9.71184 

0.28816  1.9416 

45 

l6 

813  9.66099 

24 
25 

888  9.94885 

540  9.71215 

0.28785   402 

44 

17 

839  9.66124 

875  9.94878 

7 

577  9.71246 

0.28754   388 

43 

l8 

865  9.66148 

24 

862  9.94871 

7 

6 

7 
6 

614  9.71277 

0.28723   375 

42 

19 

891  9.66173 

25 
24 

848  9-94865 

651  9.71308 

31 

31 
30 
31 
31 
31 
31 
31 
31 
31 
31 

0.28692   361 

41 

20 

45917  9.66197 

88835  9-94858 

51688  9.71339 

0.28661  1.9347 

40 

21 

942  9.66221 

24 
25 

822  9.94852 

724  9.71370 

0.28630   333 

39 

22 

968  9.66246 

808  9-94845 

7 
6 

761  9.71401 

0.28599   319 

38 

23 

994  9.66270 

24 

795  9-94839 

798  9.71431 

0.28569   306 

37 

24 

46020  9.66295 

25 
24 

782  9-94832 

7 
6 

835  9.71462 

0.28538   292 

3^ 

25 

46046  9.66319 

88768  9.94826 

51872  9.71493 

0.28507  1.9278 

35 

26 

072  9.66343 

24 
25 

755  9.94819 

7 

6 

909  9^71524 

0.28476   265 

34 

27 

097  9.66368 

741  9-94813 

946  9.71555 

0.28445   251 

33 

28 

123  9.66392 

24 

728  9.94806 

7 

983  9.71586 

0.28414   237 

32 

29 

149  9.66416 

24 

25 

715  9.94799 

7 
6 

52020  9.71617 

0.28383   223 

31 

30 

46175  9.66441 

88701  9.94793 

52057  9.71648 

0.28352  1.9210 

30 

31 

201  9.66465 

24 

688  9.94786 

I 

094  9.71679 

0.28321   196 

29 

32 

226  9.66489 

24 

674  9.94780 

131  9.71709 

30 
31 
31 
31 
31 

0.28291   183 

28 

33 

252  9.66513 

24 

661  9-94773 

7 
6 

7 

168  9.71740 

0.28260   169 

27 

34 

278  9-66537 

24 
25 

647  9-94767 

205  9.71771 

0.28229   155 

26 

35 

46304  9.66562 

88634  9.94760 

52242  9.71802 

0.28198  1.9142 

25 

36 

330  9.66586 

24 

620  9.94753 

I 

279  9.71833 

0.28167   128 

24 

37 

355  9.66610 

24 

607  9.94747 

316  9.71863 

30 

0.28137   115 

23 

38 

381  9.66634 

24 

593  9-94740 

7 
6 

7 

353  9.71894 

31 
31 
30 
31 
31 
31 
30 
31 
31 
30 
31 
30 
31 
31 
30 
31 
30 
31 
30 
31 
30 
3^ 
30 

0.28106   lOI 

22 

39 

407  9.66658 

24 
24 

580  9-94734 

390  971925 

0.28075   088 

21 

40 

46433  9.66682 

88566  9.94727 

52427  9.71955 

0.28045  1.9074 

20 

41 

458  9.66706 

24 
25 

553  9-94720 

I 

464  9.71986 

0.28014   061 

19 

42 

484  9.66731 

539  9-94714 

501  9.72017 

0.27983   047 

18 

43 

510  9.66755 

24 

526  9-94707 

7 

538  9.72048 

0.27952   034 

17 

44 

536  9.66779 

24 
24 

512  9.94700 

7 
6 

575  9.72078 

0.27922   020 

16 

45 

46561  9.66803 

88499  9.94694 

52613  9.72109 

0.27891  1.9007 

15 

46 

587  9.66827 

24 

485  9-94687 

7 

650  9.72140 

0.27860  1.8993 

14 

47 

613  9.66851 

24 

472  9.94680 

7 
6 

687  9.72170 

0.27830   980 

i3 

48 

639  9.66875 

24 

458  9.94674 

724  9.72201 

0.27799   967 

12 

49 

664  9-66899 

24 
23 

445  9.94667 

7 
7 
6 

761  9.72231 

0.27769   953 

II 

50 

46690  9.66922 

88431  9.94660 

52798  9.72262 

0.27738  1.8940 

10 

=;i 

716  9.66946 

24 

417  9.94654 

836  9.72293 

0.27707   927 

9 

f^s 

742  9.66970 

24 

404  9.94647 

7 

873  9.72323 

0.27677   913 

8 

';3 

767  9.66994 

24 

390  9.94640 

I 

7 
I 

910  9.72354 

0.27646   900 

7 

^4 

793  9.67018 

24 
24 

377  9.94634 

947  9-72384 

0.27616   887 

6 

55 

46819  9.67042 

88363  9.94627 

52985  9-72415 

0.27585  1.8873 

5 

■;6 

844  9.67066 

24 

349  9.94620 

53022  9.72445 

0.27555   860 

4 

^^7 

~  870  9.67090 

24 

336  9.94614 

059  9.72476 

0.27524   847 

3 

'^S 

896  9.671 13 

23 

322  9.94607 

7 

096  9.72506 

0.27494   834 

2 

I'o 

921  9.67137 

24 

308  9.94600 

7 

134  9.72537 

0.27463   820 

I 

947  9.67161 

24 

295  9-94593 

7 

171  9.72567 

0.27433   807 

0 

JNatCoSLog.  d. 

Nat.  Sin  Log.  d. 

Nat.  Cot  Log 

c.d 

Log.TanNat 

□ 

62^ 


Nat.  Sin  Log.    d. 


2_8^ 

Nat.  Cos  Log.  d. 


Nat.TanLog.  c.d.  Log.  Cot  Nat 


46947 
973 
999 

47024 
050 


9.67161 
9.67185 
9.67208 
9.67232 
9.67256 


47076 

lOI 

127 

153 

178 


9.67280 

9-67303 
9.67327 

967350 
9-67374 


47204 
229 

255 
281 
306 


9.67398 
9.67421 

9-67445 
9.67468 
9.67492 


47332 
358 
383 
409 

434 


9-67515 
9-67539 
9.67562 
9.67586 
9.67609 


47460 
486 
511 
537 
562 


9-67633 
9-67656 
9.67680 
9.67703 
9.67726 


47588 
614 

639 
665 
690 


9.67750 

9-67773 
9.67796 
9.67820 
9.67843 


47716 
741 
767 

793 
818 


9.67866 
9.67890 
9.67913 
9.67936 
9.67959 


47844 
869 

895 
920 
946 


9.67982 
9.68006 
9.68029 
9.68052 
9.68075 


47971 

997 

48022 

048 

073 


9.68098 
9.68121 
9.68144 
9.68167 
9.68190 


48099 
124 
150 
175 
201 


9.68213 
9.68237 
9.68260 
9.68283 
9.68305 


252 
277 
303 
328 


9.68328 
9-68351 
9-68374 
9.68397 
9.68420 


48354 
379 
405 
430 
456 
481 


9.68443 
9.68466 
9.68489 
9.68512 

9-68534 
9.68557 


88295 
281 
267 

254 
240 


9-94593 
9-94587 
9.94580 

9-94573 
9-94567 


88226 
213 
199 
185 
172 


9.94560 
9-94553 
994546 
9-94540 
9-94533 


88158 
144 
130 
117 
103 


9.94526 
994519 
994513 
9-94506 
9-94499 


075 
062 
048 
034 


9.94492 
9.94485 
9.94479 
9.94472 
9-94465 


88020 
006 

87993 
979 
965 


9.94458 
9-94451 
9.94445 
9.94438 

9.94431 


87951 
937 
923 
909 


9.94424 
9.94417 
9.94410 
9-94404 
9-94397 


7882 
868 

854 
840 
826 


9-94390 
9.94383 
9-94376 
9-94369 
9.94362 


87812 
798 
784 
770 
756 


9-94355 
9-94349 
9-9434? 
9-94335 
9.94328 


87743 
729 

715 
701 
687 


9.94321 
9-94314 
9-94307 
9-94300 
9-94293 


87673 
659 
645 
631 
617 


9.94286 
9.94279 

9-94273 
9.94266 

9-94259 


87603 
589 
575 

546 


9.94252 

994245 
9.94238 
9.94231 
9-94224 


87532 
518 

504 
490 
476 

462 


9.94217 
9.94210 
9-94203 
9.94196 
9.94189 
9.94182 


53171 
208 
246 
283 
320 


9-72567 
9.72598 
9.72628 

9-72659 
9.72689 


53358 
395 
432 
470 
507 


9.72720 
9.72750 
9.72780 
9.7281 1 
9.72841 


53545 
582 
620 

657 
694 


9.72872 
9.72902 
9.72932 
9-72963 
9.72993 


53732 
769 
807 

844 
882 


9-73023 

9-73054 
9.83084 

9-73114 
9-73144 


53920 
957 
995 

54032 
070 


9-73175 
9-7320$ 
9-7323$ 
9-7326$ 
9-73295 


54107 
145 
183 
220 
258 


973326 
9-73356 
9-73^6 
9.73416 
9-73446 


54296 
333 
371 
409 
446 


9-73476 
9-73507 
9-73537 
9-73567 
9-73597 


54484 
522 
560 
597 
635 


9-73627 
9-73657 
9-73687 
9-73717 
9-73747 


54673 
711 
748 
786 
824 


9-73777 
9.73807 
9.73837 
9.73867 

9-73897 


54862 
900 
938 
975 

55013 


9.73927 
9-73957 
9-73987 
9.74017 

9-74047 


55051 
089 
127 

165 
203 


9-74077 
9.74107 

9-74137 
9.74166 
9-74196 


55241 
279 
317 

355 
393 
431 


9.74226 
9-74256 
9.74286 
9.74316 
9.7434$ 
974375 


0.27433 
0.27402 
0.27372 
0.27341 
0.27311 


794 
781 
768 
755 


0.27280 
0.27250 
0.27220 
0.27189 
0-27159 


1.8741 
728 

715 
702 
689 


0.27128 
0.27098 
0.27068 
0.27037 
0.27007 


.8676 
663 
650 

637 
624 


0.26977 
0.26946 
0.26916 
0.26886 
0.26856 


1.8611 
598 
585 
572 
559 


0.2682$ 
0.26795 
0.26765 
0.26735 
0.26705 


1.8546 

533 
520 

507 
495 


0.26674 
0.26644 
0.26614 
0.26584 
0.26554 


1.8482 
469 
456 
443 
430 


0.26524 
0.26493 
0.26463 
0.26433 
0.26403 


1.8418 
405 
392 
379 
367 


0.26373 
0.26343 
0.26313 
0.26283 
0.26253 


0.26223 
0.26193 
0.26I63 
0.26133 
0.26103 


1-8354 
341 
329 
316 

303 
1.8291 

278 
265 

253 
240 


0.26073 
0.26043 
0.26013 
0.25983 
0.25953 


1.8228 

^  215 

202 

190 

177 


0.25923 
0.25893 
0.25863 
0.25834 
0.25804 


1.8 165 
152 
140 
127 
115 


0.25774 
0.25744 
0.25714 
0.25684 
0.25655 
0.2562.5 


1.8 103 
090 
078 
065 

053 

040 


Nat. Cos  Log.  d.   Nat.  Sin  Log.   d.  Nat.  Cot  Log. | c.d.  Log.Tan  Nat.|   ^ 

61° 


29° 

Nat.  Sin  Log.  d.  Nat.  Cos  Log.  d.  Nat.Tan  Log.  c.d 


Log.  Cot  Nat. 


48481 
506 

532 
557 
5B3 


9-68557 
a.68580 
0.68603 
9.68625 
9.68648 


48608 
634 
659 
684 
710 


9.68671 
9.68694 
9.68716 
9.68739 
9.68762 


48735 
761 
786 
811 
837 


9.68784 
9.68807 
9.68829 
9.68852 
9.68875 


913 
938 
964 


9.68897 
9.68920 
9.68942 
9.68965 
9.68987 


48989 

49014 

040 

065 

090 


9.69010 
9.69032 

969055 
9.69077 
9.69100 


491 16 
141 
166 
192 
217 


9.69122 
9.69144 
9.69167 
9.69189 
9.69212 


49242 
268 

293 
318 
344 


9.69234 
9.69256 
9.69279 
9.69301 
9-69323 


49369 
394 
419 

445 
470 


9-69345 
9.69368 
9.69390 
9.69412 
969434 


49495 
521 
546 
571 
596 


9.69456 
9.69479 
9.69501 
9.69523 
9-69545 


49622 
647 
672 
697 
723 


9.69567 
9.69589 
9.6961 1 

9-69633 
9-69655 


49748 
773 
798 
824 
849 


9.69677 
9.69699 
9.69721 

9-69743 
9.69765 


49874 
899 
924 
950 
975 

50000 


9.69787 
9.69809 

969831 
9.69853 
9.69875 
9.69297 


87462 
448 

434 
420 
406 


9.94182 

9.94175 
9.94168 
9.94161 
9.94154 


87391 
377 
363 
349 
335 


9.94147 
9.94140 

9.94133 
9.94126 

9-94119 


87321 
306 
292 
278 
264 


9.941 12 
9.94105 
9.94098 
9.94090 
9.94083 


87250 

235 
221 
207 
193 


9.94076 
9.94069 
9.94062 

9-94055 
9.94048 


87178 
164 

150 
136 

121 


9.94041 

9-94034 
9.94027 
9.94020 
9.94012 


87107 

^3 
079 
064 
050 


9.94005 
9.93998 
9-93991 
9-93984 
9-93977 


87036 
021 
007 

86993 
978 


9-93970 
9-93963 
9-93955 
993948 
9.93941 


86964 
949 
935 
921 
906 


9.93934 
9-93927 
9-93920 
9.93912 
9.93905 


86892 
878 
863 
849 
834 


9.93898 
9.93891 
9.93884 
9.93876 
9.93869 


86820 
805 
791 

m 
762 


9.93862 
9.93855 
9-93847 
9.93840 

9-93833 


86748 

733 
719 
704 
690 


9.93826 
9.93819 
9.9381 1 
9-93804 

9-93797 


86675 
661 
646 
632 
617 
603 


993789 
9.93782 

9-93775 
9.93768 
9-93760 
9-93753 


55431 
469 

507 
545 

583 


9-74375 
9-74405 
9-74435 
9.74465 
9.74494 


55621 
659 
697 
736 
774 


974524 
9.74554 
9-74583 
9.74613 
9.74643 


55812 
8:;o 


964 


9.74673 
9.74702 

9.74732 
9-74762 
9.74791 


56003 
041 
079 
117 
156 


9.74821 
9.74851 
9.74880 
9.74910 
9.74939 


56194 
232 
270 
309 
347 


9.74969 
9.74998 
9.75028 

9.75058 
9.75087 


56385 
424 
462 

501 
539 


9.75"7 
9.75146 
9.75176 
9.75205 
9.75235 


56577 
616 

654 
693 
731 


9.75264 
9-75294 
9-75323 
9.75353 
9-75382 


56769 
808 
846 
885 
923 


9-754" 
9-75441 
9-75470 
9-75500 
975529 


56962 

57000 

039 

078 

116 


9.75558 
9.75588 
9.75617 
9.75647 
9.75676 


57155 
193 
232 
271 
309 


9.75705 
9-75735 
9-75764 
9-75793 
9-75822 


57348 
386 
425 
464 
503 


9.75852 
9.75881 
9.75910 
9.75939 
9.75969 


57541 
580 
619 

657 

696 


9.75998 
9.76027 
9.76056 
9.76086 
9.761 15 
9.76144 


0.25625 

0.2559$ 
0.2556$ 

0.25535 
0.25506 


1.8040 
028 
016 
003 

1.7991 


0.25476 
0.25446 
0.25417 
0.25387 
0.25357 


1.7979 
966 

954 
942 

930 


0.25327 
0.25298 
0.25268 
0.25238 
0.25209 


1.7917 
905 
893 
881 


0.25179 
0.25149 
0.25120 
0.25090 
0.25061 


1.7856 
844 
832 
820 
808 


0.25031 
0.25002 
0.24972 
0.24942 
0.24913 


1.7796 

783 
771 

759 
747 


0.24883 
0.24854 
0.24824 
0.2479$ 
0.24765 


1-7735 
723 
711 

699 
687 


0.24736 
0.24706 
0.24677 
0.24647 
0.24618 


1.7675 
663 

651 
639 
627 


0.24589 

0.24559 
o.24$30 
0.24500 
0.24471 


1.7615 
603 
591 

579 
567 


0.24442 
0.24412 
0.24383 

0.24353 
0.24324 


1.7556 
544 
532 
520 
508 


0.24295 
0.2426^ 
0.24236 
0.24207 
0.24178 


1.7496 
485 
473 
461 

449 


0.24148 
0.241 19 
0.24090 
0.24061 
0.24031 


1.7437 
426 
414 
402 
391 


0.24002 

0.23973 
0.23944 
0.23914 
0.23885 
0.23856 


1-7379 
367 
355 
344 
332 
321 


Nat.CoSLog.  d.  Nat.  Sin  Log.  d.  Nat.CotLog.  c.d.lLog.TanNat 


30 

0 

r 

Nat.  Sin  Log.  d. 

Nat.  Cos  Log.  d. 

Nat.TanLog. 

c.d. 

Log.  Cot  Nat. 

0 

50000  9.69897 

86603  9-93753 

57735  9-76144 

0.23856  1.7321 

60 

I 

02s  9.69919 

22 

588  9.93746 

8 

774  9-76173 

29 
29 

0.23827   309 

59 

2 

050  9.69941 

573  9-93738 

813  9.76202 

0.23798   297 

58 

3 

076  9.69963 

559  9.93731 

7 

851  9.76231 

29 
30 
29 

29 
29 
29 
29 
29 
29 
29 

0.23769   286 

57 

4 
5 

loi  9.69984 

22 

544  9-93724 

7 
7 

Q 

890  9.76261 

0.23739   274 

56 

50126  9.70006 

86530  9-93717 

57929  9.76290 

0.23710  1.7262 

55 

6 

151  9.70028 

515  9-93709 

7 

I 

968  9-76319 

0.23681   251 

54 

7 

176  9.70050 

501  9.93702 

58007  9.76348 

0.23652   239 

S3 

8 

201  9.70072 

486  9.93695 

046  9-76377 

0.23623   228 

52 

9 

227  9.70093 

22 

471  9-93687 

7 

7 
8 

085  9.76406 

0.23594   216 

SI 
50 

10 

50252  9.701 15 

86457  9.93680 

58124  9.76435 

0.23565  1.7205 

11 

277  9-70137 

442  9-9367$ 

162  9.76464 

0.23536   193 

49 

12 

302  9.70159 

427  9.93665 

201  9.76493 

0.23507   182 

48 

13 

327  9.70180 

413  9-93658 

8 

240  9.76522 

29 
29 

0.23478   170 

47 

14 

352  9.70202 

22 

398  9-93650 

7 
I 

279  9-76551 

0.23449   159 

46 

15 

50377  9.70224 

86384  9-93643 

58318  9.76580 

0.23420  1.7147 

45 

16 

403  9.76245 

369  9.93636 

357  9-76609 

0.23391   13b 

44 

17 

428  9.70267 

354  9.93628 

7 
7 
8 

7 
8 

396  9-76639 

30 
29 
29 
28 
29 
29 

0.23361   124 

43 

i8 

453  9.70288 

340  9.93621 

435  9-76668 

0.23332   113 

42 

19 

478  9.70310 

22 

325  9.93614 

474  9-76697 

0.23303   102 

41 

20 

50503  9.70332 

86310  9.93606 

58513  9-76725 

0.23275  1.7090 

40 

21 

528  9-70353 

295  9-93599 

552  9-76754 

0.23246   079 

39 

22 

553  9-70375 

28  T  9.93591 

591  9-76783 

0.23217   067 

38 

23 

578  9.70396 

266  9.93584 

7 
8 

631  9.76812 

29 
29 
29 
29 

29 
29 
29 

29 
29 

^8 

0.23188   056 

.37 

24 

603  9.70418 

21 

251  9-93577 

670  9.76841 

0.23159   045 

36 

25 

50628  9.70439 

86237  9-93569 

58709  9.76870 

0.23130  1.7033 

35 

26 

654  9.70461 

222  9.93562 

7 
8 

748  9-76899 

0.23101   022 

34 

27 

679  9.70482 

207  9-93554 

7 

8 

787  9.76928 

0.23072   on 

33 

28 

704  9.70504 

192  9-93547 

826  9-76957 

0.23043  1.6999 

32 

29 

729  9.70525 

22 

178  9-93539 

7 

865  9.76986 

0.23014   988 

31 
30 

30 

50754  9.70547 

86163  9.93532 

58905  9.77015 

0.22985  1.6977 

31 

779  9.70568 

148  9.93525 

8 

944  9-77044 

0.22956   965 

29 

32 

804  9.70590 

133  9-93517 

983  9-77073 

0.22927   954 

28 

33 

829  9.7061 1 

119  9-93510 

8 

59022  9.77101 

29 
29 

0.22899   943 

27 

34 

854  9-70633 

21 

104  9-93502 

7 
8 

061  9.77130 

0.22870   932 

26 
25 

35 

50879  9.70654 

86089  9.93495 

59101  9.77159 

0.22841  1.6920 

36 

904  9.70675 

074  9.93487 

140  9.77188  1  z 

0.22812   90Q 

24 

37 

929  9.70697 

21 

059  993480 

7 
8 

179  9.-772I7 

29 

0.22783   898 

23 

3« 

954  9-70718 

21 

045  993472 

218  9.77246 

0.22754   887 

22 

39 
"40 

979  9-70739 

22 

030  9-93465 

8 

258  9.77274 

29 

29 
29 

29 

28 

0.22726   875 

21 

51004  9.70761 

86015  9-93457 

59297  9-77303 

0.22697  1.6864 

20 

41 

029  9.70782 

000  9-93450 

8 

336  9-77332 

0.22668   853 

19 

42 

054  9.70803 

85985  9-93442 

376  9-77361 

0.22639   842 

18 

43 

079  9.70824 

970  9.93435 

8 

415  9-77390 

0.22610   831 

17 

44 

104  9.70846 

21 

956  9-93427 

7 
8 

454  9-77418 

29 
29 
29 

'^8 

0.22582   820 

16 

45 

51 129  9.70867 

85941  9-93420 

59494  9-77447 

0.22553  1.6808 

15 

46 

154  9-70888 

926  9.93412 

7 
8 

533  9-77476 

0.22524   797 

14 

47 

179  9-70909 

911  9.93405 

573  9-77505 

0.22495   786 

13 

48 

204  9.70931 

896  9-93397 

612  9.77533 

29 
29 

"8 

0.22467   775 

12 

49 
50 

229  9.70952 

21 
21 

881  9.93390 

7 
8 

7 
8 

651  9.77562 

0.22438   764 

11 

51254  9.70973 

85866  9.93382 

59691  9.77591 

0.22409  1.6753 

10 

51 

279  9-70994 

851  9-93375 

730  9.77619 

29 
29 
29 
28 

29 

08 

0.22381   742 

9 

S2 

304  9-71015 

836  9-93367 

770  9.77648 

0.22352   731 

8 

53 

329  9.71036 

821  9.93360 

8 

809  9.77677 

0.22323   720 

7 

54 
55 

354  9-71058 

21 

806  9-93352 

8 
7 

849  9-77706 

0.22294   709 

b 
5 

51379  9-71079 

85792  9-93344 

59888  9.77734 

0.22266  1.6698 

56 

404  9.71 100 

777  9-93337 

928  9.77763 

0.22237   687 

4 

57 

429  9.71 121 

762  993329 

967  9.77791 

29 
29 

28 

0.22209   676 

3 

5» 

454  9.71 142 

747  9-93322 

8 

60007  9.77820 

0.22180   665 

2 

^0 

479  9-71 163 

732  9-93314 

7 

046  9.77849 

0.22151   654 

1 

504  9.71184 

1  717  9-93307 

086  9.77877 

0.22123   643 

0 

Nat.CoSLog.  d.  |Nat.  Sin  Log.  d. 

Nat.  Cot  Log. 

C.d. 

Log.TanNat. 

f 

m 


31° 

f 

Nat.  Sin  Log.  d. 

Nat.  Cos  Log.  d. 

Nat.TanLog. 

c.d. 

Log.  Cot  Nat. 

0 

51504  9.71184  1  ,^ 

85717  9-93307 

8 

60086  9.77877 

29 

0.22123  1.6643 

60 

I 

529  9.71205 

702  9.93299 

8 

126  9.77906 

0.22094   632 

SQ 

2 

554  9-71226 

21 

687  9-93291 

I 

165  9-77935 

29 
28 

0.22065   621 

58 

3 

579  9-71247 

672  9.93284 

205  9-77963 

0.22037   610 
0.22008   599 

57 

4 

604  9.71268 

21 

657  9-93276 

7 
8 

245  9-77992 

29 
28 
29 

56 

65 

5 

51628  9.71289 

85642  9-93269 

60284  9.78020 

0.21980  1.6588 

b 

653  9-71310 

627  9.93261 

8 

324  9-78049 

0.21951   577 

54 

7 

678  9-71331 

612  9.93253 

7 

8 

364  9-78077 

29 
29 

0.21923   566 

53 

8 

703  9-71352 

597  9-93246 

403  9.78106 

0.21894   555 

52 

9 

728  9-71373 

20 
21 

582  9.93238 

8 

7 

8 

443  9-78135 

0-21865   545 

51 

10 

51753  9.71393 

85567  9-93230 

60483  9.78163 

29 

28 

0.21837  1.6534 
0.21808   523 

50 

II 

778  9-71414 

551  9-93223 

522  9.78192 

49 

12 

803  9-71435  21 

536  9-93215 

0 

562  9.78220 

29 

28 

0.21780   512 

48 

13 

828  9.71456  f. 

521  9.93207 

7 
8 

Q 

602  9.78249 

0.21751   501 

47 

14 

852  9.71477 

21 
21 

506  9.93200 

642  9.78277 

- 

29 

08 

0.21723   490 

46 
45 

15 

51877  9.71498 

85491  9-93192 

60681  9.78306 

0.21694  1.6479 

lb 

902  9.71519 

20 

476  9-93184 

721  9.78334 

28 

0.21666   469 

44 

17 

927  9-71539 

21 

461  9-93177 

0 

761  9.78363 

0.21637   458 

43 

i8 

952  9.71560 

21 

446  9.93169 

0 

801  9.78391 

08 

0.21609   447 

42 

19 

977  9-71581 

21 
20 

431  9.93161 

7 

Q 

841  9-78419 

29 

0.21581   436 

41 

20 

52002  9.71602 

85416  9-93154 

60881  9.78448 

0.21552  1.6426 

40 

21 

026  9.71622 

21 

401  9.93146 

Q 

921  9.78476 

29 

0.21524   415 

39 

22 

051  9-71643 

21 

385  9-93138 

960  9-78505 

0.21495   404 

38 

23 

076  9.71664 

21 

370  9.93131 

0 

61000  9.78533 

29 
28 

0.21467   393 

37 

24 

loi  9.71685 

355  9-93123 

8 

7 

8 

040  9.78562 

0.21438   383 

3^ 

52126  9-71705 

'>T 

85340  9-93115 

61080  9.78590 

O.21410  1.6372 

35 

26 

151  9.71726 

325  9.93108 

120  9.78618 

29 

0.21382   361 

34 

27 

175  9-71747 

310  9.93100 

8 

160  9.78647 

0.21353   351 

33 

28 

200  9.71767 

^1 

294  9-93092 

0 

200  9.78675 

29 

28 

'^8 

0.21325   340 

32 

29 

225  9.71788 

21 
20 

279  9.93084 

7 
0 

240  9.78704 

0.21296   329 

31 
30 

30 

52250  9.71809 

85264  9-93077 

61280  9.78732 

0.21268  1.6319 

31 

275  9.71829 

21 

249  993069 

Q 

320  9.78760 

29 

.0.21240   308 

29 

32 

299  9-71850 

20 

234  9-93061 

0 

360  9.78789 

O.21211   297 

28 

33 

324  9.71870 

218  9.93053 

7 
8 

8 

400  9.788i'7 

^^8 

O.21183   287 

27 

34 

349  9-71891 

20 
21 

203  9-93046 

440  9.78845 

29 

08 

0.21 155   276 

26 

35 

52374  9-71911 

85188  9.93038 

61480  9.78874 

O.21126  1.6265 

25 

3b 

399  9-71932 

173  9-93030 

8 

520  9.78902 

'>8 

0.21098   25s 

24 

37 

423  9.71952 

157  9-93022 

8 

561  9-78930 

29 

'>8 

0.21070   244 

23 

3» 

448  9.71973 

142  9.93014 

601  9.78959 
641  9-78987 

O.21041   234 

22 

39 

473  9-71994 

20 

127  9.93007 

7 
8 
8 

28 

08 

O.21013   223 

21 

40 

52498  9.72014 

85112  9-92999 

61681  9.79015 

0.20985  1.6212 

20 

41 

522  9.72034 

21 

096  9-92991 

0 

721  9.79043 

29 

o9 

0.20957   202 

19 

42 

547  9-7205$ 

081  9-92983 

761  9-79072 

0.20928   191 

18 

43 

572  9-72075 

21 

066  9.92976 

0 

801  9.79100 

o<? 

0.20900   181 

17 

44 

597  9-72096 

21 

051  9.92968 

8 

Q 

842  9.79128 

28 

29 

08 

0.20872   170 

16 

45 

52621  9.721 16 

85035  9.92960 

61882  9.79156 

0.20844  1.6160 

15 

46 

646  9.72137 

020  9.92952 

8 

922  9-79185 

0.20815   149 

14 

47 

671  9.72157 

005  9.92944 

0 

962  9.79213 

o<^ 

0.20787   139 

i3 

48 

696  9.72177 

21 

84989  9.92936 

62003  979241 

08 

0.20759   128 

12 

49 

720  9.72198 

20 

974  9-92929 

8 
8 

043  9-79269 

28 
29 

-^8 

0.20731    118 

II 

50 

52745  9.72218 

84959  9-92921 

62083  9.79297 

0.20703  1.6107 

10 

51 

770  9-72238 

21 

943  9-92913 

8 

124  9.79326 

0.20674   097 

9 

S2 

794  9-72259 

928  9-92905 

8 
0 

164  9-79354 

28 

0.20646   087 

8 

S3 

819  9.72279 

913  9.92897 

204  9.79382 

'^8 

0.20618   076 

7 

54 

844  9.72299 

21 

897  9-92889 

8 

245  9.79410 

28 

'>8 

0.20590   066 

6 
5 

55 

52869  9.72320 

84882  9.92881 

62285  9.79438 

0.20562  1.6055 

S6 

893  9-72340 

866  9.92874 

I 

325  9.79466 

29 

'^8 

0.20534   045 

4 

S7 

918  9.72360 

851  9.92866 

8 

366  9-79495 

0.20505   034 

3 

S» 

943  9-72381 

836  9.92858 

8 

406  9-79523 

"8 

0.20477   024 

2 

SQ 

967  9.72401 

820  9.92850 

8 

446  979551 

'^H 

0.20449   014   I  1 

60 

992  9.72421 

80s  9.92842 

487  9-79579 

0.20421   003  0 1 

Nat.  Cos  Log.  d. 

Nat.  Sin. Log.  d. 

|Nat.CotLog. 

C.d. 

Log. Tan  Nat. 

u 

68^ 


f 

Nat.  Sin  Log.  d. 

Nat.  Cos  Log.  d. 

Nat.TanLog. 

c.d. 

Log.  Cot  Nat. 

0 

52992  9.72421 

20 

84805  9.92842 

g 

62487  9.79579 

28 

0.20421  1.6003 

60 

I 

53017  9-72441 

789  9.92834 

8 

527  9.79607 

28 

0.20393  1.5993 

59 

2 

041  9.72461 

774  9-92826 

8 

568  9-79635 

28 

0.20365   983 

58 

3 

066  9.72482 

759  9.92818 

8 

608  9.79663 

-28 

0.20337   972 

57 

4 
6 

091  9.72502 

20 

743  9.92810 

7 

8 

649  9.79691 

28 

08 

0.20309   962 

56 
55 

53IIS  9.72522 

84728  9.92803 

62689  9-79719 

0.20281  1.5952 

b 

140  9.72542 

712  9.92795 

8 

730  9-79747 

29 
28 

0.20253   941 

54 

7 

164  9.72562 

" 

697  9-92787 

8 

770  9.79776 

0.20224   931 

53 

8 

189  9.72582 

^ 

681  9-92779 

8 

811  9.79804 

28 

0.20196   921 

52 

9 

214  9.72602 

20 

666  9.92771 

8 

Q 

852  9-79832 

28 
28 

0.20168   911 

51 
50 

10 

53238  9.72622 

84650  9.92763 

62892  9.79860 

0.20140  1.5900 

II 

263  9.72643 

635  992755 

Q 

933  9-79888 

28 

0.201 12   890 

49 

12 

288  9.72663 

619  9.92747 

Q 

973  9-79916 

28 

0.20084   880 

48 

13 

312  9-72683 

604  992739 

8 

63014  9.79944 

'-R 

0.20056   869 

47 

14 

337  9-72703 

20 

588  9.92731 

8 

8 

055  9-79972 

28 
"8 

0.20028   859 

4b 
45 

15 

53361  9.72723 

84573  9.92723 

63095  9.80000 

0.20000  1.5849 

lb 

386  9-72743 

557  9-92715 

0 

136  9.80028 

28 

0.19972   839 

44 

17 

411  9.72763 

542  9.92707 

8 

177  9.80056 

28 

0.19944   829 

43 

I8 

435  9-72783 

~ 

526  9.92699 

8 

217  9.80084 

28 

0.19916   818 

42 

19 

460  9.72803 

20 

511  9.92691 

8 

8 

258  9.80112 

28 

-^8 

0.19888   808 

41 
40 

20 

53484  9.72823 

84495  9.92683 

63299  9.80140 

0.19860  1.5798 

21 

509  9.72843 

480  9.92675 

Q 

340  9.80168 

27 

28 

0.19832   788 

39 

22 

534  9-72863 

464  9.92667 

8 

380  9.80195 

0.19805   778 

38 

23 

558  9.72883 

19 
20 

448  9.92659 

8 

421  9.80223 

'^8 

0.19777-   768 

37 

24 

583  9.72902 

433  9-92651 

8 

8 

462  9.80251 

28 
"8 

0.19749   757 

36 
35 

25 

53607  9.72922 

84417  9.92643 

63503  9.80279 

0.19721  1.5747 

2b 

632  9.72942 

402  9.92635 

0 

544  9-80307 

-^8 

0.19693   737 

34 

27 

656  9.72962 

386  9.92627 

8 

584  9.80335 

28 

0.19665   727 

33 

28 

681  9.72982 

370  9.92619 

8 

625  9.80363 

-^8 

0.19637   717 

32 

29 

705  9-73002 

20 
19 

355  9.92611 

8 

8 

666  9.80391 

28 
08 

0.19609   707 

31 
30 

30 

53730  9-73022 

84339  9.92603 

63707  9.80419 

0.19581  1.8697 

31 

754  973041 

324  9.92595 

0 

748  9.80447 

27 
28 

0.19553   687 

29 

32 

779  9-73061 

308  9.92587 

0 

789  9-80474 

0.19526   677 

28 

33 

804  9.73081 

292  9.92579 

9 

830  9.80502 

28 

0.19498   667 

27 

34 
35 

828  9.73101 

20 
19 

277  9-92571 

8 

8 

871  9.80530 

28 

'^8 

0.19470   657 

2b 

53853  9-73121 

84261  9.92563 

63912  9.80558 

0.19442  1-5647 

25 

3(5 

877  9-73140 

245  9-92555 

9 

8 

953  9-80586 

'^H 

0.19414   637 

24 

37 

902  9.73160 

" 

230  9-92546 

994  9.80614 

28 

0.19386   627 

23 

3» 

92b  9.73180 

20 

214  9.92538 

8 

64035  9.80642 

27 
28 
28 

0.19358   617 

22 

39 
40 

951  9.73200 

19 

198  9-92530 

8 
0 

076  9,80669 

0.19331   607 

21 
20 

53975  9-73219 

84182  9.92522 

641 17  9.80697 

0.19303  1-5597 

41 

54000  9-73239 

167  9.92514 

8 

158  9.80725 

08 

0.19275   587 

19 

42 

024  9-73259 

19 

151  9.92506 

8 

199  9-80753 

28 

0.19247   577 

18 

43 

049  9-73278 

135  9.92498 

8 

240  9.80781 

27 
28 

08 

0.19219   567 

17 

44 

073  9-73298 

20 
19 

120  9.92490 

8 

9 

8 

281  9.80808 

0.19192   557 

lb 

45 

54097  9-73318 

84104  9.92482 

64322  9.80836 

0.19164  1.5547 

15 

46 

122  9-73337 

088  9.92473 

363  9.80864 

08 

0.19136   537 

14 

47 

146  9-73357 

072  9.92465 

8 

404  9.80892 

27 

'^8 

0.19108   527 

13 

48 

171  9-73377 

19 
20 

19 

057  9-92457 

8 

446  9.80919 

0.19081   517 

12 

49 

195  9-73396 

041  9.92449 

8 

8 

487  9.80947 

28 
28 

0.19053   507 

II 

50 

54220  9-73416 

84025  9.92441 

64528  9.80975 

0.19025  1.5497 

10 

51 

244  9-73435 

009  9.92433 

8 

569  9.81003 

27 
28 

0.18997   487 

9 

S2 

269  9-73455 

" 

83994  9.92425 

9 

8 

610  9.81030 

0.18970   477 

8 

=^3 

293  9.73474 

19 

978  9.92416 

652  9.81058 

28 

0.18942   468 

7 

54 

317  9-73494 

19 

962  9.92408 

8 

8 

693  9.81086 

27 

"8 

O.18914   458 

b 

55 

54342  9-73513 

83946  9.92400 

64734  9.81 1 13 

0.18887  1.5448 

5 

,0 

366  9-73533 

930  9.92392 

0 

775  9.81 141 

28 

0.18859   438 

4 

57 

391  9-73552 

915  9.92384 

8 

817  9.81 169 

27 
28 

O.18831   428 

3 

58 

415  9-73572 

899  9-92376 

9 

0 

858  9.81 196 

0.18804   418 

2 

ro 

440  9-73591 

19 

883  9.92367 

899  9.81224 

o« 

0.18776   408 

I 

404  9-73611 

2°  1   8b7  9-92359 

941  9.81252  1  -  \  0.18748   399 

0 

Nat.  Cos  Log.  d. 

Nat.  Sin  Log.  d. 

Nat.  Cot  Log. 

c.d. 

Log.TanNat. 

f 

57' 


33° 


1 

Nat.  Sin  Log.  d. 

Nat.  Cos  Log 

d. 

Nat.TanLog. 

c.d. 

Log.  Cot  Nat. 

"^ 

0 

54464  9.7361 1 

19 

83867  9.92359 

8 

64941  9.81252 

0.18748  1.5399 

60 

I 

488  9.73630 

851  9.92351 

8 

982  9.81279 

28 

0.18721   389 

59 

2 

513  9-73650 

19 

835  9.92343 

8 

65024  9.81307 

0.18693   379 

58 

3 

537  9.73669 

819  9.92335 

9 
8 

Q 

065  9.81335 

0.18665   369 

S7 

4 

561  9.73689 

19 
19 
20 

804  9.92326 

106  9.81362 

27 

28 
28 

0.18638   359 

56 

55 

5 

54586  9.73708 

83788  9.92318 

65148  9.81390 

0.18610  1.5350 

6 

610  9.73727 

772  9.92310 

0 

189  9.8I4I8 

0.18582   340 

54 

7 

635  9.73747 

19 
19 
20 

19 
19 
20 

756  9.92302 

9 

8 

231  9.81445 

28 
27 
28 
28 

0.18555   330 

53 

8 

659  9.73766 

740  9.92293 

272  9.81473 

0.18527   320 

52 

9 

683  9.73785 

724  9.92285 

8 
8 

314  9.81500 

0.18500   311 

51 
50 

10 

54708  9.73805 

83708  9.92277 

65355  9.81528 

0.18472  1.5301 

11 

732  9.73824 

692  9.92269 

397  9.81556 

0.18444   291 

4Q 

12 

756  9.73843 

676  9.92260 

9 
8 

438  9.81583 

27 
28 

0.18417   282 

48 

13 

781  973863 

19 
19 

660  9.92252 

8 

9 
8 
8 
8 
9 
8 
8 

480  9.81611 

0.18389   272 

47 

14 

805  9.73882 

645  9.92244 

521  9.81638 

27 

28 

0.18362   262 

46 
45 

15 

54829  9.73901 

83629  9.92235 

65563  9.81666 

0.18334  1.5253 

lb 

854  9.73921 

19 
19 
19 
19 

613  9.92227 

604  9.81693 

27 

28 

0.18307   243 

44 

17 

878  9.73940 

597  9.92219 

646  9.81721 

0.18279   233 

43 

18 

902  9.73959 

581  9.9221 1 

688  9.81748 

27 

08 

0.18252   224 

42 

19 

927  9.73978 

565  9.92202 

729  9.81776 

27 
28 

0.18224   214 

41 

20 

54951  9-73997 

83549  9.92194 

65771  9.81803 

0.18197  1.5204 

40 

21 

975  9.74017 

19 
19 
19 
19 

533  9.92186 

813  9.81831 

0.18169   195 

39 

22 

999  9-74036 

517  9.92177 

9 
8 
8 

9 
8 
8 

854  9.81858 

0! 

0.18142   185 

38 

23 

55024  9.74055 

501  9.92169 

896  9.81886 

0.18114   175 

37 

24 

048  9.74074 

485  9.92161 

938  9.81913 

27 
28 

0.18087   166 

36 

25 

55072  9.74093 

83469  9.92152 

65980  9.81941 

0.18059  1.5 156 

35 

2b 

097  9.74113 

19 
19 
19 
19 
19 
19 
19 

453  9.92144 

66021  9.81968 

28 

0.18032   147 

34 

27 

121  9.74132 

437  9.92136 

063  9.81996 

0.18004   137 

33 

2b 

145  9.74151 

421  9.92127 

9 

105  9.82023 

27 

0.17977   127 

32 

29 

169  9.74170 

405  9.92119 

8 

147  9.82051 

27 

28 

0.17949   118 

31 

30 

55194  9.74189 

83389  9.921 1 1 

66189  9.82078 

0.17922  1.5108 

30 

31 

218  9.74208 

373  9.92102 

9 
8 
8 

230  9.82106 

0.17894   099 

29 

32 

242  9.74227 

356  9.92094 

272  9.82133 

s 

0.17867   089 

28 

33 

266  9.74246 

340  9.92086 

314  9.82161 

0.17839   080 

27 

34 

291  9.74265 

■••y 
19 
19 
19 
19 

324  9.92077 

9 
8 

9 

8 
0 

356  9.82188 

27 

27 

28 

0.17812   070 

26 
25 

35 

55315  9.74284 

83308  9.92069 

66398  9.82215 

0.17785  1.5061 

3(' 

339  9.74303 

•292  9.92060 

440  9.82243 

0.17757   051 

24 

37 

363  9.74322 

276  9.92052 

482  9.82270 

^8 

0.17730   042 

23 

3« 

388  9.74341 

260  9.92044 

524  9.82298 

0.17702   032 

22 

39 
40 

412  9.74360 

••^9 
19 
19 
19 
19 

244  9.92035 

9 
8 

566  9.82325 

27 
27 
28 

0.17675   023 

21 

55436  9.74379 

83228  9.92027 

66608  9.82352 

0.17648  1.5013 

20 

41 

460  9.74398 

212  9.92018 

8 
8 

650  9.82380 

0.17620   004 

19 

42 

484  9.74417 

195  9.92010 

692  9.82407 

28 

0.17593  1.4994 

18 

43 

509  9.74436 

179  9.92002 

734  9.82435 

0.17565   985 

17 

44 
45 

533  9-74455 

19 
19 

163  9.91993 

9 
8 

27 
27 
28 

0.17538   975 

lb 
15 

55557  9.74474 

83147  9.91985 

66818  9.82489 

0.17511  14966 

46 

581  9.74493 

131  9.91976 

9 
8 

860  9.82517 

27 
27 
28 

0.17483   957 

14 

47 

605  9.74512 

■^y 

115  9.91968 

902  9.82544 

0.17456   947 

13 

48 

630  9.74531 

098  9.91959 

9 
0 

944  9.82571 

0.17429   938 

12 

49 

654  9.74549 

19 
19 
19 

082  9.91951 

9 
8 

986  9.82599 

27 
27 
28 

0.17401   928 

11 
10 

50 

55678  9.74568 

83066  9.91942 

67028  9.82626 

0.17374  1.4919 

51 

702  9.74587 

050  9-91934 

071  9.82653 
113  9.82681 

0.17347   910 

9 

S2 

726  9.74606 

034  9-91925 

27 
27 
27 

0.17319   900 

8 

SS 

750  9.74625 

■••9 

017  9.91917 

155  9.82708 

0.17292   891 

7 

54 

775  9.74644 

19 

18 

001  9.91908 

9 

8 

197  9.82735 

0.17265   882 

b 

55 

55799  9.74662 

82985  9.91900 

67239  9.82762 

0.17238  14872 

5 

0 

823  9.74681 

^y 

969  9.91891 

282  9.82790 

27 
27 

0.17210   863 

4 

'57 

847  9-74700 

19 

953  9.91883 

324  9.82817 

0.17183   854 

3 

58 

871  9.74719 

i"i 

936  9.91874 

8 

366  9.82844 

0.17156   844 

2 

ro 

895  9.74737 

920  9.91866 

409  9.82871 

0.17129   835 

1 

919  9.74756 

■••9 

904  9.91857 

9 

451  9.82899 

0.17101   826 

0 

Nat.  Cos  Log.  d. 

Nat.  Sin  Log 

d. 

Nat.  Cot  Log. 

cd. 

Log.TanNat. 

/ 

56' 


34^ 


'   Nat.  Sin  Log.  d. 


NatCoSLosf.  d. 


Nat.TanLog. 


d.  Log.  Cot  Nat, 


55919 
943 
968 
992 

56016 


9-74756 
9-74775 
9.74794 
9.74812 
9.74831 


56040 
064 
088 
112 
136 


9.74850 
9.74868 
9-74887  , 

9-74906 !  g 
9-74924 


56160 
184 
208 
232 
256 


9-74943 
9.74961 
9.74980 
9.74999 
9.75017 


56280 
305 
329 
353 
377 


9-75036 
975054 
9.75073 
9.75091 
9.751 10 


56401 
425 
449 
473 
497 


9.75128 

9.75147 
9-75165 
9.75184 
9.75202 


56521 
545 
569 
593 
617 


9.75221 
975239 
9-75258 
9-75276 
9-75294 


56641 
665 
689 
713 
736 


9-75313 
9-75331 
9-75350 
9-75368 
9-75386 


56760 
784 
808 
832 
856 


9-75405 
9-75423 
9-75441 
9-75459 
9.75478 


Daao 
904 
928 
952 
976 


9.75496 
9-75514 
9-75533 
9-75551 
9-75569 


57000 
024 
047 
071 
095 


9-75587 
975605 
9-75624 
9.75642 
9-75660 


57119 
143 
167 
191 
215 


9-75678 
9.75696 
9-75714 
9-75733 
9-75751 


57238 
262 
286 
310 
334 
358 


9-75769 
9-7.5787 
9-75805 
9-75823 
9.75841 

9-75859 


82904 
887 
871 
855 
839 


9.91857 
9.91849 
9.91840 
9.91832 
9.91823 


806 
790 
773 
757 


9.91815 
9.91806 
9.91798 
9.91789 
9.91781 


8274T 
724 
708 
692 
675 


9.91772 
9.91763 

9-91755 
9.91746 
9.91738 


82659 

643 
626 
610 
593 


9.91729 
9.91720 
9.91712 
9.91703 
9.91695 


82577 
561 
544 
528 

511 


9.91686 
9.91677 
9.91669 
9.91660 
9-91651 


82495 
478 
462 
446 
429 


9.91643 
9.91634 
9.91625 
9.91617 
9.91608 


82413 
396 
380 
363 
347 


9-91599 
9.91591 
9.91582 
9-91573 
9-91565 


82330 
314 
297 
281 
264 


9-91556 
9-91547 
9-91538 
9-91530 
9.91521 


82248 
231 
214 
198 
181 


9.91512 
9.91504 

991495 
9.91486 
9.91477 


82165 
148 
132 

115 
098 


9.91469 
9.91460 
9.91451 
9.91442 
9-91433 


82082 
065 
048 
032 
oiS 


9.91425 
9.91416 
9.91407 
9.91398 
9-91389 


81999 
982 
965 
949 
932 
915 


9.91381 
9.91372 
9-91363 
9-91354 
9-91345 
9-91336 


67451 
493 
536 
578 
620 


9.82899 
9.82926 

9-82953 
9.82980 
9.83008 


67663 

705 
748 
790 
832 


9-83035 
9.83062 
9-83089 
9.83117 
9-83144 


67875 
917 
960 

68002 
045 


9.83171 
9.83198 
9.83225 
9-83252 
9.83280 


130 
173 
215 
258 


9-8.3307 
9-83334 
9-83361 
9-83388 
9.83415 


68301 

343 
386 
429 
471 


9.83442 
9.83470 
9.83497 
9.83524 
9.83551 


68514 
557 
600 
642 
685 


9-83578 
9.83605 
9.83632 

983659 
9.83686 


68728 
771 
814 

857 
900 


9-83713 
9.83740 
9.83768 

9-83795 
9.83822 


68942 

985 

69028 

071 

114 


9.83849 
9.83876 
9.83903 
9-83930 
9-83957 


69157 
200 

243 
286 

329 


9.83984 
9.84011 
9-84038 
9-84065 
9.84092 


69372 
416 

459 
502 

545 


9.84119 
9.84146 
9.84173 
9.84200 
9.84227 


69588 
631 

675 
718 
761 


9.84254 
9.84280 
9.84307 

9-84334 
9.84361 


69804 

847 
891 

934 

977 

70021 


9.84388 
9.84415 
9.84442 
9.84469 
9.84496 
9-84523 


7101  1.4826 

7074  816 

7047  807 

7020  798 

6992  788 


6965  1-4779 
6938  770 
6911  761 
6883  751 
6856   742 


6829  1.4733 


6802 
6775 
6748 
6720 


724 
715 
705 
696 


6693  1.4687 
6666  678 
6639  669 
6612  659 
6585   650 


6558  1.4641 
6530  632 
6503  623 
6476  614 
6449   605 


6422  1.4596 

6395  586 

6368  577 

6341  568 

6314  559 


6287  1.4550 
6260  541 
6232  532 
6205  523 
6178   514 


6151  1.4505 
6124  496 
6097  487 
6070  478 
6043   469 


6016  1.4460 

5989   451 
5962   442 

5935   433 
5908   424 


5881  1.4415 

5854  406 

5827  397 

5800  388 

5773   379 


5746  1.4370 
5720  361 
5693  352 
5666  344 
5639   335 


5612  1.4326 
558S  317 
5558  308 
5531  299 
5504  290 
5477   281 


Nat.  Cos  Log.  d, 


Nat.  Sin  Log.    d. 

66^ 


Nat.  Cot  Log. 


c.d.  Log.TanNat. 


{ 

35 

0 

f 

Nat.  Sin  Log.  d. 

Nat.  Cos  Log 

d. 

Nat.TanLog. 

c.d.  Log.  Cot  Nat. 

0 

57358  9-75859  i  18 

81915  9.91336 

8 

70021  9.84523 

0.15477  1.428 1 

60 

I 

381  9-75877  i  18 

899  9.91328 

9 

9 

9 

9 

9 

9 
8 

064  984550 

06 

0.15450   273 

59 

2 

405  9-75895  18 

882  9.91319 

107  9.84576 

27 
27 
27 
27 
27 

0.15424   264 

58 

3 

429  9.75913  1  t8 

865  9-91310 

151  9.84603 

0.15397   255 

57 

4 

453  9-75931 

18 

848  9.91301 

194  9-84630 

0.15370   246 

56 
55 

5 

57477  9-75949 

81832  9.91292 

70238  9.84657 

0.15343  1.4237 

6 

501  9.75967 

iH 

815  9.91283 

281  9.84684 

0.15316   229 

54 

7 

524  9-75985 

18 

798  9-91274 

325  9.847II 

0.15289   220 

S3 

8 

548  9-76003 

iR 

782  9.91266 

9 
9 
9 
9 
9 
9 
9 
9 
9 

368  9-84738 

26 

0.15262   211 

S2 

9 

572  9.76021 

18 

765  9.91257 

412  9-84764 

27 
27 

0.15236   202 

51 

10 

57596  9-76039 

T« 

81748  9.91248 

70455  9.84791 

0.15209  1.4193 

II 

619  9-76057 ! ;« 

731  9.91239 

499  9-84818 

0.15182   185 

49 

12 

643  9.76075 

18 

714  9.91230 

542  9.84845 

0.15155   176 

48 

1.3 

667  9-76093 

18 

698  9.91221 

586  9.84872 

0.15128   167 

47 

14 

691  9.761 1 1 

18 

681  9.91212 

629  9.84899 

26 

27 
27 

0.15101   158 

46 
45 

15 

57715  9.76129 

TT 

81664  9-91203 

70673  9.84925 

0.15075  1.4150 

I6 

738  9.76146  ^ 

647  9.91194 

717  9-84952 

0.15048   141 

44 

17 

762  9.76164  :  8 

631  9.91185 

760  9-84979 

0.15021   132 

43 

i8 

786  9.76182 1 ;« 

614  9.91176 

9 

804  9.85006 

27 

0.14994   124 

42 

19 

810  9.76200 

18 

t8 

597  9-91167 

9 
9 

9 

8 

848  9.85033 

27 
26 

0.14967   115 

41 
40 

20 

57833  9-762i« 

81580  9.91158 

70891  9.85059 

0.14941  1.4106 

21 

857  9-76230  ,, 

563  9.91149 

935  9-85086 

0.14914   097 

39 

22 

881  9.76253 

18 

546  9.91141 

9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
10 

9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 

979  9-85113 

"7 
27 
26 

0.14887   089 

38 

23 

904  9.76271 

18 

530  9.91 132 

71023  9.85140 

0.14860   080 

37 

24 

928  9.76289 

t8 

513  9.91123 

066  9.85166 

27 
27 

27 

05 

0.14834   071 

36 
35 

25 

57952  9-76307 

18 

81496  9.91114 

71  no  9.85193 

0.14807  1.4063 

26 

976  9-76324 

479  9-91 105 

154  9.85220 

0.14780   054 

34 

27 

999  9-76342 

18 

462  9.91096 

198  9.85247 

0.14753   045 

33 

28 

58023  9.76360 

18 

445  9-91087 

242  9.85273 

0.14727-   037 

32 

29 

30 

047  9-76378 

17 
18 

428  9.91078 

285  9-85300 

27 

27 

27 

0.14700   028 

31 
30^ 

58070  9-76395 

8 14 1 2  9.91069 

71329  9.85327 

0.14673  1.4019 

31 

094  9.76413 

18 

395  9.91060 

373  9-85354 

0.14646   on 

29 

32 

118  9.76431 

;? 

378  9.91051 

417  9.85380 

0.14620   002 

28 

33 

141  9-76448 

361  9.91042 

461  9.85407 

27 
26 

27 
27 

06 

0.14593  1.3994 

27 

34 
35 

165  9.76466  1  11 

344  9-91033 

505  9-85434 

0.14566   985 

26 

58189  9.76484 

17 
18 

81327  9.91023 

71549  9.85460 

0.14540  1.3976 

25 

36 

212  9.76501 

310  9.91014 

593  9-85487 

0.14513   968 

24 

37 

236  9-76519 

18 

293  9.91005 

637  9-85514 

0.14486   959 

23 

38 

260  9-76537 

681  9-85540 

0.14460   951 

22 

39 
40 

283  9-76554 

17 
18 
18 

259  9.90987 

725  9.85567 

27 

27 

06 

0.14433   942 

21 
20 

58307  9-76572 

81242  9.90978 

71769  9.85594 

0.14406  1.3934 

41 

330  9-76590 

17 

18 

225  9.90969 

813  9.85620 

0.14380   925 

19 

42 

354  9-76607 

208  9.90960 

857  9-85647 

27 

0.14353   916 

18 

43 

378  9.76625 

191  9.90951 

901  9.85674 

06 

0.14326   908 

17 

44 

401  9.76642 

18 

174  9-90942 

946  9.85700 

27 
27 

^6 

0.14300   899 

16 

45 

58425  9.76660 

81 157  9-90933 

71990  9.85727 

0.14273  1.3891 

15 

46 

449  9-76677 

140  9.90924 

72034  9.85754 

0.14246   882 

14 

47 

472  9.76695 

123  9.90915 

078  9.85780 

0.14220   874 

13 

48 

496  9.76712 

^8 

17 
18 

106  9.90906 

122  9.85807 

0.14193   865 

12 

49 
50 

519  9-76730 

089  9.90896 

9 
9 
9 
9 
9 
9 

167  9-85834 

26 

0.14166   857 

11 

58543  9-76747 

81072  9.90887 

72211  9.85860 

0.14140  1.3848 

10 

■^i 

567  9.76765 

055  990878 

255  9-85887 

0.14113   840 

9 

S2 

590  9.76782 

18 

038  9.90869 

299  9-85913 

27 
27 
26 

27 

0.14087   831 

8 

';3 

614  9.76800 

021  9.90860 

344  985940 

0.14060   823 

7 

54 
55 

637  9.76817 

17 
18 

004  9.90851 

388  9-85967 

0.14033   814 

6 

58661  9.76835 

80987  9.90842 

72432  9.85993 

0.14007  1.3806 

5 

S6 

684  9.76852 

17 
18 

970  9.90832 

9 

477  9.86020 

0.13980   798 

4 

708  9.76870 

953  990823 

521  9.86046 

27 
27 
'>6 

0.13954   789 

3 

58 

731  9.76887 

17 

936  9.90814 

9 
9 

565  9.86073 

0.13927   781 

2 

i 

755  9-76904 

18 

919  9.90805 

610  9.86100 

0.13900   772 

I 

779  9-76922 

902  9-90796 

9 

654  9.86126 

0.13874   764 

0 

Nat.CoSLog.  d. 

Nat.  Sin  Log. 

d. 

Nat.  Cot  Log. 

c.d. 

Log.Tan  Nat. 

f 

64 

t° 

36 

0 

t 

Nat.  Sin  Log.  d. 

Nat.  Cos  Log.  d. 

Nat. Tan  Log. 

c.d. 

Log.  Cot  Nat. 

0 

58779  9-76922 

80902  9.90796 

72654  9.86126 

0.13874  1.3764 

60 

I 

802  9.76939 

iS 

885  9.90787 

699  9.86153 

06 

0.13847   755 

59 

2 

826  9.76957 

17 

867  9.90777 

743  9-86179 

0.13821   747 

58 

3 

849  9-76974 

850  9.90768 

788  9.86206 

26 
27 

'^6 

0.13794   739 

S7 

4 

873  9.76991 

17 
18 

17 
17 

18 

833  9.90759 

9 
9 
9 

832  9.86232 

0.13768   730 

56 

5 

58896  9.77009 

80816  9.90750 

72877  9-86259 

0.13741  1.3722 

55 

b 

920  9.77026 

799  9.90741 

921  9.86285 

% 

0.13715   713 

54 

7 

943  9-77043 

782  9.90731 

966  9.86312 

0.13688   705 

53 

b 

967  9.77061 

17 
17 
17 
18 

765  9.90722 

9 
9 

73010  9.86338 

0.13662   697 

52 

9 

990  9.77078 

748  9.90713 

055  9-86365 

27 

0.13635   688 

51 

10 

59014  9-77095 

80730  9-90704 

73100  9.86392 

0.13608  1.3680 

50 

II 

037  9-77"2 

713  9.90694 

.  9 
9 

144  9.86418 

0.13582   672 

49 

12 

061  9-77130 

17 

696  9.90685 

189  9.86445 

o(S 

0.13555   663 

48 

13 

084  9.77147 

679  9.90676 

234  9.86471 

0.13529   655 

47 

14 

108  9.77164 

17 
17 

18 

662  9.90667 

9 
10 

9 

278  9.86498 

27 
26 

0.13502   647 

46 

15 

S9131  9.77181 

80644  9-90657 

73323  9.86524 

0.13476  1.3638 

45 

16 

154  9-77199 

627  9.90648 

368  9.86551 

26 
05 

0.13449   630 

44 

17 

178  9.77216 

17 
17 
18 

17 
17 
17 

610  9.90639 

9 
9 

413  9-86577 

0.13423   622 

43 

l8 

201  9.77233 

593  9.90630 

457  9.86603 

0.13397   613 

42 

19 

225  9-77250 

576  9.90620 

9 

502  9.86630 

26 

0.13370   605 

41 

20 

59248  9.77268 

80558  9.90611 

73547  9-86656 

0.13344  1.3597 

40 

21 

272  9.77285 

541  9.90602 

"6 

0.13317   588 

39 

22 

295  9-77302 

524  9.90592 

9 

637  9.86709 

27 

0.13291   580 

38 

23 

318  9-77319 

507  9.90583 

681  9.86736 

0.13264   572 

37 

24 

342  9.77336 

17 
17 
17 
17 

489  9.90574 

9 
9 

726  9.86762 

27 
05 

0.13238   564 

36 

25 

59365  9-77353 

80472  9.90565 

73771  9.86789 

0.13211  1.3555 

35 

2b 

389  9-77370 

455  9.90555 

816  9.86815 

0.13185   547 

34 

27 

412  9.77387 

438  9.90546 

9 
9 

861  9.86842 

o(=, 

0.13158   539 

33 

28 

436  9.77405 

17 
17 
17 
17 
17 

420  9.90537 

906  9.86868 

'^f, 

0.13132   531 

32 

29 

459  9-77422 

403  9.90527 

9 
9 

951  9.86894 

27 

2*^ 

0.13106   522 

31 

30 

59482  9.77439 

80386  9.90518 

73996  9.86921 

0.13079  1.3514 

30 

31 

506  9.77456 

368  9.90509 

74041  9.86947 

% 

0.13053   506 

29 

32 

529  9.77473 

351  9.90499 

9 

086  9.86974 

0.13026   498 

28 

33 

552  9-77490 

334  9-90490 

131  9.87000 

27 
26 

"6 

0.13000   490 

27 

34 
35 

576  9-77507 

17 
17 
17 
17 
17 
17 
17 

316  9.90480 

9 

176  9.87027 

0.12973   481 

2b 

59599  9-77524 

80299  9.90471 

74221  9-87053 

p.12947  1.3473 

25 

3^^ 

622  9.77541 

282  9.90462 

9 

267  9.87079 

0.12921   465 

24 

37 

646  9-7755? 

264  9.90452 

312  9.87106 

06 

0.12894   457 

23 

3« 

669  9.77575 

247  990443 

9 

9 

10 

357  9-87132 

^(^ 

0.12868   449 

22 

39 
40 

693  9.77592 

230  9.90434 

402  9.87158 

27 

06 

0.12842   440 

21 

59716  9.77609 

80212  9.90424 

74447  9-87185 

0.12815  1.3432 

20 

41 

739  9.77626 

195  9-90415 

9 

492  9.87211 

0.12789   424 

19 

42 

763  9.77643 

17 

178  9-90405 

538  9.87238 

27 
26 

0.12762   416 

18 

43 

786  9.77660 

TT 

160  9.90396 

583  9.87264 

06 

0.12736   408 

17 

44 

809  9.77677 

17 

143  9-90386 

9 

628  9.87290 

27 
06 

0.12710   400 

lb 

45 

59832  9.77694 

80125  9.90377 

74674  9.87317 

0.12683  1.3392 

16 

4b 

856  9.77711 

17 

108  9.90368 

9 

719  9.87343 

06 

0.12657   384 

14 

47 

879  9.77728 

16 

091  9-90358 

764  9-87369 

27 

06 

0.12631   375 

13 

48 

902  9.77744 

073  9-90349 

9 

810  9.87396 

0.12604   367 

12 

49 

926  9.77761 

17 
17 
17 
17 

056  9-90339 

9 

855  9-87422 

26 
27 

0.12578   359 

11 

50 

59949  9-77778 

80038  9.90330 

74900  9.87448 

0.12552  1.3351 

10 

51 

972  9.77795 

021  9.90320 

9 

946  9.87475 

0.12525   343 

9 

.S2 

995  9-77812 

003  9.9031 1 

991  9.87501 

06 

0.12499   335 

8 

S3 

60019  9.77829 

79986  9.90301 

75037  9-87527 

27 
26 

■2(^ 

0.12473   327 

7 

54 

042  9.77846 

16 

968  9.90292 

10 

082  9-87554 

0.12446   319 

b 

55 

60065  9.77862 

79951  9.90282 

75128  9.87580 

0.12420  1. 33 II 

5 

St 

089  9.77879 

17 

934  9-90273 

9 

173  9.87606 

% 

0.12394   303 

4 

57 

112  9.77896 

17 

916  9.90263 

219  9-87633 

0.12367   295 

3 

5« 

135  9.77913 

17 

899  9-90254 

9 

264  9-87659 

26 

0.12341   287 

2 

;'J?. 

158  9.77930 

881  9.90244 

310  9.87685 

26 

0.12315   278 

I 

60 

182  9.77946 

864  9.90235 

9 

355  9-8771 1 

0.12289   270 

0 

Nat.  Cos  Log.  d. 

Nat.  Sin  Log.  d. 

Nat.  Cot  Log. 

C.d. 

Log.TanNat. 

lI 

63^ 


Nat.  Sin  Log.  d. 


37 

Nat.  Cos  Log.  d. 


Nat.TanLog.  c.d 


Log.  Cot  Nat, 


35 

36 
37 
38 

40 

41 
42 

43 

44 


60182 
205 
228 

251 
274 


9.77946 

977963 
9.77980 
9.77997 
9.78013 


60298 
321 
344 
367 
390 


9.78030 
9.78047 
9.78063 
9.78080 
978097 


60414 

437 
460 

483 
506 


9.781 13 
9.78130 
9.78147 
9.78163 
9.78180 


60529 
553 
576 
599 
622 


9.78197 
9.78213 
9.78230 
9.78246 
9.78263 


60645 
668 
691 
714 
738 


9.78280 
9.78296 

9.78313 
9.78329 
9.78346 


60761 
784 
807 
830 
853 


9.78362 
978379 
978395 
9.78412 
9.78428 


60876 
899 
922 

945 
968 


9-78445 
9.78461 
9.78478 
9.78494 
9.78510 


60991 

61015 

038 

061 

084 


9.78527 

9-78543 
9.78560 
9.78576 
9.78592 


61107 
130 

176 
199 


9.78609 
9.78625 
9.78642 
9.78658 
9-78674 


61222 

245 
268 
291 
314 


9.78691 
9.78707 
9.78723 

9-78739 
9.78756 


61337 
360 

383 
406 
429 


9.78772 
9.78788 
9.78805 
9.78821 
9.78837 


6145 1 
474 
497 
520 

543 
566 


9-78853 
9.78869 
9.78886 
9.78902 
9.78918 
9.78934 


79864 
846 
829 
811 
793 


9.9023.5 
9.90225 
9.90216 
9.90206 
9.90197 


79776 
758 
741 

723 
706 


9.90187 
9.90178 
9.90168 
9.90159 
9.90149 


79688 
671 
653 
635 
618 


9.90139 
9.90130 
9.90120 
9.90111 
9.90101 


79600 

583 
565 
547 
530 


9.90091 
9.90082 
9.90072 
9.90063 
9.90053 


79512 
494 
477 
459 
441 


9-90043 
9.90034 
9.90024 
9.90014 
9.90005 


79424 
406 


371 

353 


9-8999$ 
9.89985 
9.89976 
9.89966 
9.89956 


79335 
318 
300 
282 
264 


9.89947 

9-89937 
9.89927 
9.89918 
9.89908 


79247 
229 
211 

193 
176 


9.89898 
9.89888 

9.89879 
9.89869 

9.89859 


79158 
140 
122 

105 

087 


9.89849 
9.89840 
9.89830 
9.89820 
9.89810 


79069 
051 
033 
016 

78998 


9.89801 
9.89791 
9.89781 
9.89771 
9.89761 


78980 
962 

944 
926 
908 


9.89752 
9.89742 
9.89732 
9.89722 
9.89712 


78891 
873 
855 
837 
819 
801 


9.89702 
9.89693 
9.89683 

9.89673 
9.89663 

9.89653 


75355 
401 

447 
492 
538 


9.8771 1 
9.87738 

9.87764 
9.87790 
9.87817 


75584 
629 

675 
721 
767 


9.87843 
9.87869 
9.87895 
9.87922 
9.87948 


75812 
858 
904 

950 
996 


9.87974 
9.88000 
9.88027 
9.88053 
9.88079 


76042 
088 

134 
180 
226 


9.88105 
9.88131 
9.88158 
9.88184 
9.88210 


76272 
318 
364 
410 
456 


9.88236 
9.88262 
9.88289 

9.88315 
9.88341 


76502 
548 
594 
640' 
686 


9.88367 
988393 
9.88420 
9.88446 
9.88472 


76733 
779 
825 
871 
918 


9.88498 
9.88524 
9.88550 
9.88577 
9.88603 


76964 
77010 

057 
103 
149 


9.88629 
9.88655 
9.88681 
9.88707 
9.88733 


77196 
242 
289 

335 
382 


9.88759 
9.88786 
9.88812 
9.88838 
9.88864 


77428 
475 
521 
568 
615 


9.88890 
9.88916 
9.88942 
9.88968 
9.88994 


77661 
708 

754 
801 


9.89020 
9.89046 
9.89073 
9.89099 
9.89125 


77895 
941 
988 

78035 
082 
129 


9.89151 
9.89177 
9.89203 
9.89229 

989255 
9.89281 


Nat.  Sin  Log.    d.  Nat.  Cot  Log.  c.d,  Log.TanNat 


0.12289 
0.12262 
0.12236 
0.12210 
0.12183 


1.3270 
262 

254 
246 
238 


0.12157 
0.12131 
0.12105 
0.12078 
0.12052 


1.3230 
222 

214 
206 
198 


0.12026 
0.12000 
0.11973 
0.11947 
0.11921 


1.3190 
182 

175 
167 

159 


0.11895 
0.11869 
0.11842 
0.11816 
0.11790 


1-3151 
143 
135 
127 
119 


0.11764 
0.11738 
0.11711 
0.11685 
0.11659 


1.3111 
103 

095 
087 
079 


0.11633 
0.11607 
0.11580 

0.11554 
0.11528 


1.3072 
064 
056 
048 
040 


0.11502 
0.11476 
0.11450 
0.11423 
0.11397 


1.3032 
024 
017 
009 
001 


0.11371 

0.11345 
0.11319 
0.11293 
0.11267 


1.2993 
985 
977 
970 
962 


0.11241 
0.11214 
0.11188 
0.11162 
0.1 1 136 


1.2954 
946 
938 
931 
923 


O.IIIIO 

0.11084 
0.11058 
0.11032 
0.11006 


1.2915 
907 
900 
892 


0.10980 
0.10954 
0.10927 
0.10901 
0.10875 


1.2876 
869 
861 

853 
846 


0.10849 
0.10823 
0.10797 
0.10771 
0.1074^ 
0.10719 


I.2fc 


830 
822 

815 
807 

799 


Nat.  Cos  Log.  d 


62^ 


38 

0 

f 

Nat.  Sin  Log.  d. 

Nat.  Cos  Log.  d. 

Nat.TanLog. 

c.d. 

Log.  Cot  Nat. 

0 

61566  9.78934 

16 

78801  9.89653 

78129  9.89281 

05 

0.10719  1.2799 

60 

I 

589  9.78950 

17 
16 

783  9-89643 

TO 

175  9.89307 

06 

0.10693   792 

59 

2 

612  9.78967 

765  9-89633 

222  9-89333 

06 

0.10667   784 

58 

3 

635  9-78983 

16 

747  9-89624 

9 

269  9.89359 

06 

0.10641   776 

^7 

4 

658  9.78999 

16 
t6 

729  9.89614 

10 

316  9-89385 

26 

26 

0.10615   769 

56 

5 

6I68I  9.79015 

787 1 1  9.89604 

78363  9.894II 

0.10589  1.2761 

55 

b 

704  979031 

16 

694  9.89594 

410  9-89437 

og 

0.10563   753 

54 

7 

726  9-79047 

16 

676  9.89584 

457  9-89463 

26 

0.10537   746 

53 

8 

749  9.79063 

16 

658  9-89574 

504  9.89489 

06 

0.10511   738 

52 

9 
10 

772  9.79079 

16 
16 

640  9-89564 

10 

551  9.89515 

26 

"6 

0.10485   731 

51 
50 

61795  9-79095 

78622  9.89554 

78598  9.89541 

0.10459  1.2723 

II 

818  9.791 1 1 

17 
16 

604  9-89544 

645  9.89567 

og 

0.10433   715 

49 

12 

841  9.79128 

586  9.89534 

692  9.89593 

06 

0.10407   708 

48 

13 

864  9.79144 

16 

568  9.89524 

739  9.89619 

og 

0.10381   700 

47 

14 

887  9.79160 

16 
t6 

550  9.89514 

10 

9 

786  9.89645 

26 

0.10355   693 

46 

15 

61909  9.79176 

78532  9.89504 

78834  9.89671 

0.10329  1.2685 

45 

lb 

932  9.79192 

16 

514  9.89495 

881  9.89697 

Ofi 

0.10303   677 

44 

17 

955  9-79208 

16 

496  9.89485 

928  9.89723 

og 

0.10277   670 

43 

l8 

978  9.79224 

16 

478  9.89475 

975  9.89749 

og 

0.10251   662 

42 

19 

62001  9.79240 

16 
16 

460  9.89465 

10 

79022  9.89775 

26 

'^6 

0.10225   655 

41 
40 

20 

62024  9.79256 

78442  9.89455 

79070  9.89801 

0.10199  1.26^ 
0.10173  -  "^ 

21 

046  9.79272 

16 

424  9.89445 

TO 

117  9.89827 

og 

39 

22 

069  9.79288 

16 

405  9.89435 

TO 

164  9.89853 

'-'6 

0.10147   632 

38 

23 

092  9.79304 

15 
16 
16 

387  9.89425 

TO 

212  9.89879 

og 

0.I0I2I    624 

37 

24 

115  9.79319 

369  9.89415 

10 

259  9.89905 

26 

26 

0.10095    617 

36 

25 

62138  9.79335 

78351  9.89405 

79306  9.89931 

0.10069  1.2609 

35 

2b 

160  9.79351 

16 

333  9.89395 

354  9.89957 
401  9.89983 

26 

0.10043    602 

34 

27 

183  9-79367 

16 

315  9.89385 

-^6 

O.IOOI7    594 

33 

28 

206  9.79383 

16 

297  9.89375 

J  J 

449  9-90009 

^f) 

0.09991    587 

32 

29 

229  9.79399 

16 
t6 

279  9.89364 

10 

496  9-90035 

26 

0^ 

0.09965   579 

31 

30 

62251  9.79415 

78261  9.89354 

79544  9.90061 

0.09939  1.2572 

30 

31 

274  9.79431 

16 

243  9.89344 

591  9.90086 

0.09914   564 

29 

32 

297  9.79447 

16 

225  9.89334 

639  9-90112 

og 

0.09888   557 

28 

33 

320  9.79463 

15 
16 
16 

206  9.89324 

686  9.90138 

-""^ 

0.09862   549 

27 

34 
35^ 

342  9.79478 

188  9.89314 

10 

734  9.90164 

26 

-^6 

0.09836   542 

2b 

25 

62365  9.79494 

78170  9-89304 

79781  9.90190 

0.09810  1.2534 

3b 

388  9.79510 

16 

152  9.89294 

TO 

829  9.90216 

^f) 

0.09784   527 

24 

37 

411  9.79526 

16 

134  9.89284 

TO 

877  9.90242 

^(^ 

0.09758   519 

23 

3H 

433  9.79542 

16 

116  9.89274 

TO 

924  9.90268 

og 

0.09732   512 

22 

39 

456  9.79558 

15 
t6 

098  9.89264 

10 

972  9-90294 

26 

og 

0.09706   504 

21 

40 

62479  9.79573 

78079  9.89254 

80020  9.90320 

0.09680  1.2497 

20 

41 

502  9.79589 

t6 

061  9.89244 

067  9.90346 

% 

0.09654   489 

19 

42 

524  9.79605 

16 

043  989233 

115  9-90371 

0.09629   482 

18 

43 

547  9.79621 

15 
16 
16 

025  9.89223 

163  9.90397 

"6 

0.09603   475 

17 

44 

570  9.79636 

007  9.89213 

10 

211  9.90423 

26 
"6 

0.09577   467 

lb 

45 

62592  9.79652 

77988  9.89203 

80258  9-90449 

0.09551  1.2460 

15 

4b 

615  9.79668 

16 

970  9.89193 

10 

306  9-90475 

'^(^ 

0.09525   452 

14 

47 

638  9.79684 

952  9.89183 

354  9-90501 

'>6 

0.09499   445 

13 

48 

660  9.79699 

934  9.89173 

402  9-90527 

'>6 

0.09473   437 

12 

49 

683  9.79715 

16 

916  9.89162 

10 

450  9-90553 

25 

0.09447   430 

II 

50 

62706  9.79731 

77897  9.89152 

80498  9.90578 

0.09422  1.2423 

10 

51 

728  9.79746 

879  9.89142 

546  9.90604 

og 

0.09396   415 

9 

52 

751  9.79762 

16 

861  9.89132 

594  9.90630 

og 

0.09370   408 

8 

53 

774  9-79778 

15 
16 
16 
IS 

16 

16 

843  9.89122 

642  9.90656 

og 

0.09344   401 

7 

54 

796  9-79793 

824  9.891 12 

II 

690  9.90682 

26 

26 

0.09318   393 

6 

~5" 

55 

62819  9.79809 

77806  9.89101 

80738  9.90708 

0.09292  1.2386 

56 

842  9.79825 

788  9.89091 

786  9.90734 

25 

26 

0.09266   378 

4 

57 

864  9.79840 

769  9.89081 

TO 

834  9.90759 
882  9.90785 

0.09241   371 

3 

5B 

887  9-79856 

751  9.89071 

26 

0.0921S   364 

2 

ro 

909  9.79872 

15 

733  9-89060 

TO 

930  9.9081 I 

og 

0.09189   356 

I 

932  9-79887 

715  9-89050 

978  9.90837 

0.09163   349 

0 

Nat.  Cos  Log.  d. 

Nat.  Sin  Log.  d. 

Nat.  Cot  Log. 

c.d. 

Log.TanNat. 

/ 

61" 


i 


39^ 


■^ 

Nat.  Sin  Log.  d.  1 

Nat.  Cos  Log.  d.| 

Nat.TanLog. 

=.d.  Log.CotNat.| 

0 

62932  9-79887 

16 

77715  9-89050 

80978  9-90837 

05 

0.09163  1.2349 

60 

I 

955  9-79903 

15 

696  9.89040 

81027  9-90863 

^f) 

0-09137  342 

59 

2 

977  9-79918 

678  9.89030 

075  9.90889 

S 

0.091 1 1   334 

58 

3 

63000  9.79934 

16 

660  9.89020 

123  9.90914 

0.09086   327 

.57 

4 

022  9.79950 

15 

16 

641  9.89009 

171  9-90940 

26 

0.09060   320 

56 

5 

63045  9.79965 

77623  9.88999 

81220  9.90966 

0.09034  1.2312 

55 

6 

068  9.79981 

15 

268  9-90992 

05 

0.09008   305 

54 

7 

090  979996 

586  9.88978 

316  9.9IOI8 

25 
06 

0.08982   298 

53 

8 

113  9.80012 

15 
16 
15 

568  9-88968 

364  9.91043 

0.08957   290 

52 

9 
10 

135  9.80027 

550  9.88958 

413  9.91069 

26 

26 

0.08931   283 

51 
50 

63158  9.80043 

77531  9.88948 

8 146 I  9.91095 

0.08905  1.2276 

II 

180  9.80058 

513  9-88937 

510  9.91 121 

26 

0.08879   268 

49 

12 

203  9.80074 

15 

494  9.88927 

558  9.91 147 

25 

26 

0.08853   261 

48 

i.S 

225  9.80089 

476  9.88917 

606  9.91 172 

0.08828   254 

47 

14 

248  9,80105 

15 
16 

458  9.88906 

655  9.9II98 

26 

0.08802   247 

46 
45 

15 

63271  9.80120 

77439  9.88896 

81703  9.91224 

0.08776  1.2239 

i6 

293  9.80136 

15 
15 
16 

421  9.88886 

752  9.91250 

nfS 

0.08750   232 

44 

I? 

316  9.80151 

402  9.88875 

800  9.91276 

% 

0.08724   225 

43 

i8 

338  9.80166 

384  9-88861 

849  9.9I30I 

0.08699   218 

42 

19 

361  9.80182 

15 

366  9-88855 

898  9-91327 

26 

0.08673   210 

41 

20 

63383  9.80197 

77347  9-88844 

81946  9-91353 

0.08647  1.2203 

40 

21 

406  9.80213 

15 

329  9.88834 

995  9-91379 

% 

0.08621    196 

39 

22 

428  9.80228 

310  9.88824 

82044  9-91404 

0.08596   189 

38 

23 

451  9.80244 

292  9.88813 

092  9.91430 

^6 

0.08570   181 

37 

24 

473  9-80259 

15 
15 

273  9.88803 

141  9.91456 

26 

0.08544   174 

36 

25 

63496  9.80274 

77255  9.88793 

82190  9.91482 

0.08518  1.2167 

35 

26 

518  9.80290 

15 

236  9.88782 

238  9-91507 

0.08493   160 

34 

27 

540  9.80305 

218  9.88772 

287  9-91533 

■^f) 

0.08467   153 

33 

28 

563  9.80320 

199  9.88761 

336  9-91559 

'^f) 

0.08441    145 

32 

29 
30 

585  9-80336 

15 

IS 

181  9.88751 

385  9-91585 

25 

0.08415    138 

31 

63608  9.80351 

77162  9.88741 

82434  9.91610 

0.08390  1.2131 

30 

31 

630  9.80366 

144  9.88730 

483  9.91636 

->f> 

0.08364    124 

29 

32 

653  9-80382 

15 
15 
16 

15 
15 

125  9.88720 

531  9.91662 

^f) 

0.08338   117 

28 

33 

675  9-80397 

107  9.88709 

580  9.91688 

25 
26 

'>6 

0.08312   109 

27 

34 

698  9.80412 

088  9.88699 

629  9.91713 

0.08287   102 

2b 
25 

35 

63720  9.80428 

77070  9.88688 

82678  9.91739 

0.08261  1.2095 

36 

742  9.80443 

051  9.88678 

727  9.91765 

06 

0.08235   088 

24 

37 

765  9.80458 

033  9.88668 

776  9.91791 

S 

0.08209   081 

23 

38 

787  9-80473 

014  9.88657 

10 

825  9.91816 

0.08184   074 

22 

39 

810  9.80489 

15 
15 
15 

76996  9.88647 

874  9.91842 

26 

25 
-^6 

0.08158   066 

21 

40 

63832  9.80504 

76977  9.88636 

82923  9.91868 

0.08132  1.2059 

20 

41 

854  9.80519 

959  9.88626 

972  9.91893 

0.08107   052 

19 

42 

877  9-80534 

940  9.88615 

83022  9.91919 

06 

0.08081   "045 

18 

43 

899  9-80550 

15 
15 
15 
15 
15 
16 

15 
15 

921  9.88605 

071  9.91945 

"6 

0.08055   038 

17 

44 

922  9.80565 

903  9.88594 

120  9.91971 

25 
'^6 

0.08029   031 

16 

45 

63944  9.80580 

76884  '9.88584 

83169  9.91996 

0.08004  1.2024 

15 

46 

966  9.80595 

866  9.88573 

218  9.92022 

-^6 

0.07978   017 

14 

47 

989  9.80610 

847  9.88563 

268  9.92048 

25 

0.07952   009 

13 

48 

64011  9.80625 

828  9.88552 

317  9.92073 

0.07927   002 

12 

49 

033  9.80641 

810  9.88542 

366  9.92099 

26 

0.07901  1.1995 

II 

50 

64056  9.80656 

76791  9.88531 

83415  9.92125 

0.07875  1. 1988 

10 

SI 

078  9.80671 

772  9.88521 

465  9.92150 

0.07850   981 

9 

q2 

100  9.80686 

15 

754  9.88510 

514  9.92176 

26 

0.07824   974 

8 

S3 

123  9.80701 

15 

735  9-88499 

564  9.92202 

25 
26 
'>6 

0.07798   967 

7 

54 
55 

145  9.80716 

15 
15 

717  9.88489 

613  9.92227 

0.07773   960 

6 

64167  9.80731 

76698  9.88478 

83662  9.92253 

0.07747  I-1953 

S6 

190  9.80746 

16 

679  9.88468 

712  9.92279 

"6 

0.07721   946 

4 

S7 

212  9.80762 

661  9-88457 

761  9.92304 

0.07696   939 

3 

S8 

234  9.80777 

15 

642  9.88447 

811  9.92330 

06 

0.07670   932 

2 

IS 

256  9.80792 

15 

623  9.88436 

860  9.92356 

25 

0.07644   925 

I 

279  9.80807 

^5 

604  9-88425 

910  9.92381 

0.07619   918 

0 

_ 

|Nat.  Cos  Log.  d. 

|Nat.  Sin  Log.  d. 

Nat.  Cot  Log 

cd 

.Log.TanNat 

/ 

60^ 


Nat.  Sin  Log.  d. 


40° 

Nat.  Cos  Log.  d.  Nat. Tan  Log 


c.d. 


Log.  Cot  Nat. 


64279 
301 
323 
346 
368 


9.80807 
9.80822 
9-80837 
9.80852 
9.80867 


64390 
412 

435 
457 
479 


9.80882 
9.80897 
9.80912 
9.80927 
9.80942 


64501 

524 
546 
568 
590 


9.80957 
9.80972 
9.80987 
9.81002 
9.81017 


64612 
635 
657 
679 
701 


9.81032 
9.81047 
9.81061 
9.81076 
9.81091 


64723 
746 
768 
790 
812 


9.81 106 
9.81121 
9.81136 
9.81151 
9.81166 


64834 
856 
878 
901 
923 


9.81180 
9.81 195 
9.81210 
9.81225 
9.81240 


64945 
967 
989 

6501 1 
033 


9.81254 
9.81269 
9.81284 
9.81299 
9-81314 


65055 
077 
100 
122 
144 


9.81328 
9-81343 
9-81358 
9.81372 
9.81387 


65166 
188 
210 
232 
254 


9.81402 
9.81417 
9.81431 
9.81446 
9.81461 


65276 
298 
320 
342 
364 


9.81475 
9.81490 
9.81505 
9.81519 
9.81534 


65386 
408 
430 
452 
474 


9.81549 
9.81563 
9.81578 
9.81592 
9.81607 


65496 
518 
540 
562 

584 
606 


9.81622 
9.81636 
9.81651 
9.81665 
9.81680 
9.81694 


76604 
586 
567 
548 
530 


9.88425 

9-88415 
9.88404 
9.88394 
9.88383 


765 1 1 
492 
473 
455 
436 


9.88372 
9.88362 

9.88351 
9.88340 
9.88330 


76417 
398 
380 
361 
342 


9.88319 
9.88308 
9.88298 
9.88287 
9.88276 


76323 
304 
286 
267 
248 


9.88266 
9.88255 
9.88244 
9-88234 
9.88223 


76229 
210 
192 
173 
154 


9.88212 
9.88201 
9.88191 
9.88180 
9.88169 


76135 
116 
097 
078 
059 


9.88158 
9.88148 

9-88137 
9.88126 
9.88115 


76041 
022 
003 

75984 
965 


9.88105 
9.88094 
9.88083 
9.88072 
9.88061 


75946 
927 
908 
889 
870 


9.88051 
9.88040 
9.88029 
9.88018 
9.88007 


75851 
832 

813 

794 
775 


9.87996 
9.87985 

987975 
9.87964 

9.87953 


75756 
738 
719 
700 
680 


9.87942 

9.87931 
9.87920 
9.87909 
9.87898 


75661 
642 
623 
604 
585 


9.87887 
9.87877 
9.87866 

9.87855 
9.87844 


75566 
547 
528 

509 
490 
471 


987833 
9.87822 
9.87811 
9.87800 
9.87789 
9.87778 


83910 
960 

84009 
059 
108 


9.92381 
9.92407 

9-92433 
9.92458 
9.92484 


84158 
208 
258 
307 
357 


9.92510 

9.92535 
9.92561 
9.92587 
9.92612 


84407 
457 
507 
556 
606 


9.92638 
9.92663 
9.92689 
9.92715 
9.92740 


84656 
706 
756 
806 
856 


9.92766 
9.92792 
9.92817 

9.92843 
9.92868 


84906 

956 

85006 

057 
107 


9.92894 
9.92920 

9-92945 
9.92971 
9.92996 


85157 
207 

257 
308 
358 


9.93022 
9.93048 

9.93073 
9.93099 
9.93124 


85408 
458 
509 
559 
609 


9-93150 
993175 
9-93201 
9-93227 
9-93252 


5660 
710 
761 
811 
862 


9.93278 
9.93303 
9.93329 
9.93354 
9.93380 


85912 
963 

86014 
064 
"5 


9.93406 
9.93431 
9-93457 
9.93482 
993508 


86166 
216 
267 
318 
368 


9-93533 
9-93559 
993584 
9.93610 
9.93636 


86419 
470 
521 
572 
623 


9.93661 
9.93687 
9.93712 
9.93738 
993763 


86674 
725 
776 
827 
878 
929 


9.93789 
9.93814 
9.93840 
993865 
9.93891 
9.93916 


26 


0.07619 

0.07593 
0.07567 
0.07542 
0.07516 


1.1915 
910 

903 
896 


0.07490 
0.07465 
0.07439 
0.07413 
0.07388 


1.1882 

875 
868 
861 
854 


0,07362 

0.07337 
0.07311 
0.07285 
0.07260 


1.1847 
840 

833 
826 
819 


0.07234 
0.07208 
0.07183 
0.07157 
0.07132 


1.1812 
806 
799 
792 
785 


0.07106 
0.07080 
0.07055 
0.07029 
0.07004 


1.1778 
771 
764 
757 
750 


0.06978 
0.06952 
0.06927 
0.06901 
0.06876 


1.1743 
736 
729 
722 
715 


0.06850 
0.06825 
0.06799 
0.06773 
0.06748 


1. 1708 
702 

695 
688 
681 


0.06722 
0.06697 
0.06671 
0.06646 
0.06620 


1. 1674 
667 
660 
653 
647 


0.06594 
0.06569 
0.06543 
0.06518 
0.06492 


1. 1 640 

633 
626 
619 
612 


0.06467 
0.06441 
0.06416 
0.06390 
0.06364 


1.1606 
599 
592 
585 
578 


0.06339 
0.06313 
0.06288 
0.06262 
0.06237 


1.1571 

565 
558 
551 
544 


0.0621 1 
0.06186 
0.06160 
o.o6i3g 
0.06109 
0.06084 


1.1538 
531 
524 
517 
510 
504 


Nat.  Cos  Log.  d.  Nat.  Sin  Log.    d.  Nat.  Cot  Log.  c.d. 

49° 


Log.TanNat. 


41 


/ 

Nat.  Sin  Log.  d. 

Nat.  Cos  Log.  d.| 

Nat.TanLog. 

c.d. 

Log.  Cot  Nat. 

0 

65606  9.81694 

15 
14 
15 

75471  9-87778 

TT 

86929  9.93916 

--•6 

0.06084  1-1504 

60 

I 

628  9.81709 

452  9-87767 

980  9.93942 

0.06058   497 

59 

2 

650  9.81723 

433  9-87756 

TT 

87031  9.93967 

0.06033   490 

58 

3 

672  9.81738 

414  9-87745 

082  9.93993 

25 
26 

0.06007   483 

57 

4 

694  9.81752 

14 
15 
14 
15 
14 
15 
14 
15 
14 
14 
15 
14 
15 
14 
15 
14 
14 

395  9-87734 

II 

133  9.94018 

0.05982   477 

56 
55 

5 

65716  9.81767 

75375  9-87723 

87184  9.94044 

0.05956  1. 1470 

6 

738  9.8I78I 

356  9.87712 

236  9.94069 

0.05931   463 

54 

7 

759  9-81796 

337  9-87701 

287  9.94095 

25 
'^6 

0.05905   456 

53 

8 

781  9.81810 

318  9.87690 

338  9.94120 

0.05880   450 

52 

9 

803  9.81825 

299  9.87679 

II 

389  9.94146 

25 

26 

0.05854   443 

51 

10 

65825  9.81839 

75280  9.87668 

87441  9.94I7I 

0.05829  1.1436 

50 

II 

847  9.81854 

261  9.87657 

492  9.94197 

2^ 

0.05803   430 

49 

12 

869  9.81868 

241  9.87646 

543  9.94222 

0.05778   423 

48 

13 

891  9.81882 

222  9.87635 

595  994248 

25 
26 

0.05752   416 

47 

14 
15 

913  9.81897 

203  9.87624 

II 

646  9.94273 

0.05727   410 

46 
45" 

65935  9-81911 

75184  9.87613 

87698  9.94299 

0.05701  1. 1403 

16 

956  9.81926 

165  9.87601 

749  994324 

0.05676   396 

44 

17 

978  9.81940 

146  9.87590 

801  9.94350 

% 

0.05650   389 

43 

l8 

66000  9.81955 

852  9.94375 

0.05625   383 

42 

19 

022  9.81969 

107  9.87568 

II 
TT 

904  9.94401 

25 

2fS 

0.05599   376 

41 

20 

66044  9-81983 

75088  9-87557 

87955  9.94426 

0.05574  1.1369 

40 

21 

066  9.81998 

14 
14 
15 
14 
14 
15 
14 
14 
14 
15 
14 
14 
15 
14 
14 
14 

069  9-87546 

88007  9.94452 

% 

0.05548   363 

39 

22 

088  9.82012 

050  9.87535 

059  9-94477 

0.05523   356 

38 

2S 

109  9.82026 

030  9.87524 

no  9.94503 

25 
26 

25 

0.05497   349 

37 

24 

25 

131  9.82041 

on  9.87513 

12 

162  9.94528 

0.05472   343 

36 

66153  9-82055 

74992  9.87501 

88214  9.94554 

0.05446  1.1336 

35 

26 

175  9.82069 

973  9.87490 

265  9-94579 

0.05421   329 

34 

27 

197  9.82084 

953  9.87479 

317  994604 

0.05396   323 

33 

28 

218  9.82098 

934  9.87468 

369  994630 

25 
26 

0.05370   316 

32 

29 

240  9.821 12 

915  9.87457 

II 

421  9.94655 

0.05345   310 

31 
30 

30 

66262  9.82126 

74896  9.87446 

88473  9.94681 

0.05319  1.1303 

31 

284  9.82141 

876  9.87434 

524  9.94706 

0.05294   296 

29 

32 

306  9.82155 

857  9.87423 

576  9.94732 

25 
26 

0.05268   290 

28 

33 

327  9.82169 

838  9.87412 

628  9.94757 

0.05243   283 

27 

34 

349  9.82184 

818  9.87401 

II 

680  9.94783 
88732  9.94808 

25 
06 

0.05217   276 

26 

25 

35 

66371  9.82198 

74799  9.87390 

0.05192  1,1270 

36 

393  9-82212 

780  9.87378 

784  9.94834 

25 

0.05166   263 

24 

37 

414  9.82226 

760  9.87367 

836  9.94859 

O.05141   257 

23 

38 

436  9.82240 

15 
14 
14 
14 
14 
15 
14 
14 

741  9.87356 

J  J 

888  9.94884 

0.051 16   250 

22 

39 

458  9-82255 

722  9.87345 

II 

940  9.94910 

25 

06 

0.05090   243 

21 

40 

66480  9.82269 

74703  9.87334 

88992  9.94935 

0.05065  1. 1 237 

20 

41 

501  9.82283 

683  9.87322 

89045  9.94961 

s 

0.05039   230 

19 

42 

523  9.82297 

664  9.87311 

097  9.94986 

0.05014   224 

18 

43 

545  9-8231 1 

644  9.87300 

149  9.95012 

25 
25 

'^6 

0.04988   217 

17 

44 

566  9.82326 

625  9.87288 

II 

201  9.95037 

0.0403   211 

16 
T5 

45 

66588  9.82340 

74606  9.87277 

89253  995062 

0.04938  1.1204 

46 

610  9.82354 

586  9.87266 

306  9.95088 

% 

0.04912   197 

H 

47 

632  9.82368 

14 

567  9.87255 

358  9.95113 

0.04887   191 

13 

48 

653  9-82382 

548  9.87243 

410  9.95139 

25 
26 

25 
25 

0.04861   184 

12 

49 
50 

675  9.82396 

14 
14 
15 

528  9.87232 

II 

463  9.95164 

0.04836   178 

II 

To 

66697  9.82410 

74509  9.87221 

89515  9.95190 

0.04810  1.1171 

=;i 

718  9.82424 

489  9.87209 

567  9.95215 

0.04785   165 

9 

=;2 

740  9.82439 

470  9.87198 

620  9.95240 

0.04760   158 

8 

S3 

762  9-82453 

14 
14 

451  9.87187 

672  9.95266 

25 
26 

% 

0.04734   152 

y 

54 

783  9.82467 

431  9.87175 

II 

725  9.95291 

0.04709   145 

6 

55 

66805  9.82481 

74412  9.87164 

89777  9.95317 

0.04683  1. 1 139 

S6 

827  9.82495 

14 

392  9.87153 

830  9.9.5342 

0.04658   132 

4 

S7 

848  9.82509 

14 

373  9.87141 

883  9.95368 

25 

0.04632   126 

3 

S8 

870  9.82523 

14 

353  9.87130 

935  9.95393 

0.04607   119 

2 

SQ 

891  9-82537 

14 

334  9.87119 

988  9.95418 

0.04582   113 

I 
0 

60 

913  9.82551 

14 

314  9.87107 

90040  9.95444 

0.04556   106 

Nat.  Cos  Log.  d. 

Nat.  Sin  Log.  d. 

Nat.  Cot  Log. 

C.d 

Log.TanNat. 

t 

m 


Nat.  Sin  Log.  d. 


42^ 

Nat.  Cos  Log.  d. 


Nat.  Tan  Log. 


c.d.  Log.  Cot  Nat, 


66913 
935 
956 
978 
999 


9.82551 
9.82565 
9.82579 

9.82593 
9.82607 


67021 

043 
064 
086 
107 


9.82621 
9.82635 
9.82649 
9.82663 
9.82677 


67129 

151 
172 
194 
215 


9.82691 
9.82705 
9.82719 
9.82733 
9.82747 


67237 
258 
280 
301 
323 


9.82761 
9.82775 
9.82788 
9.82802 
9.82816 


67344 
366 

387 
409 

430 


9.82830 
9.82844 
9.82858 
9.82872 
9.82885 


67452 
473 
495 
516 
538 


9.82899 
9.82913 
9.82927 
9.82941 
9.82955 


67559 
580 
602 
623 
645 


9.82968 
9.82982 
9.82996 
9.83010 
9.83023 


67666 
688 
709 
730 

752 


9.83037 
9.83051 
9.8306g 
9.83078 
9.83092 


67773 

795 
816 

837 
859 


9.83106 
9.83120 

9.83133 
9.83147 
9^83161 


67880 
901 
923 
944 
965 


9.83174 
9.83188 
9.83202 
9.83215 
9.83229 


67987 

68008 

029 

051 
072 


9.83242 
9.83256 
9.83270 
9.83283 
9.83297 


68093 

115 
136 

157 
179 
200 


9.83310 
9.83324 
9.83338 
9.83351 
9.83365 
9.83378 


74314 
295 
276 
256 
237 


9.87107 
9.87096 
9.87085 

9.87073 
9.87062 


74217 

9.87050 

198 

9.87039 

178 

9.87028 

159 

9,87016 

139 

9.87005 

74120 

9.86993 

100 

9.86982 

080 

9.86970 

061 

9.86959 

041 

9.86947 

74022 

9.86936 

002 

9.86924 

73983  9.86913 

963  9.86902 

944 

9.86890 

73924 

9.86879 

904 

9.86867 

885  9.86855 

865 

9.86844 

846  9.86832 

73826 

9.86821 

806 

9.86809 

787  9.86798 

767  9.86786 

747 

9.8677§ 

73728 

9.86763 

708 

9.86752 

688 

9.86740 

669  9.86728 

649  9.86717 

73629  9.86705 

610 

9.86694 

590 

9.86682 

570 

9.86670 

551 

9.86659 

73531 

9.86647 

511 

9.86635 

491 

9.86624 

472 

9.86612 

452 

9.86600 

73432 

9.86589 

413 

9.86577 

393 

9.86565 

373 

9.86554 

353 

9.86542 

73333 

9.86530 

314 

9.86518 

294 

9.86507 

274 

9.86495 

254 

9.86483 

73234 
215 
195 
175 

155 
135 


9.86472 
9.86460 
9.86448 
9.86436 
9.86425 
9.86413 


90040 

093 

146 

199 
251 


9-95444 
9.95469 

9.95495 
9.95520 
9.95545 


90304 

9.95571 

357 

410 

9.95622 

463  9.95647 

516  9.95672 

90569  9.95698 

621 

9.95723 

674  9.95748 

727 

9-95774 

781 

9.95799 

90834  9.95825 

887  9.95850 

940 

9.95875 

993 

9.95901 

91046 

9.95926 

91099 
153 

206 

259 

313 

91366 

419 

473 
526 
580 


9.95952 
9.95977 
9.96002 
9.96028 
9.96053 
9.96078 
9.96104 
9.96129 

9-96155 
9.96180 


91633 

9.96205 

687 

9.96231 

740 

9-96256 

794 

9.96281 

847  9-96307 

91901 

9.96332 

955 

92008 

9.96383 

062 

9.96408 

116 

9.96433 

92170 

9.96459 

224 

9.96484 

277 

9.96510 

331 

9.96535 

385 

9.96560 

92439 

996586 

493 

9.9661 1 

547 

996636 

601 

9.96662 

655  9.96687 

92709 

9.96712 

763  9-96738 

817  9.96763 

872  9.96788 

926  9.96814 

92980 

93034 
088 

143 

197 

252 


9.96839 

9.96864 
9.96890 

9.96915 
9.96940 

9.96966 


0.04556 
0.04531 
0.04505 
0.04480 

0.04455 


[.II06 
100 

093 
087 
080 


0.04429 
0.04404 

0.04378 
0.04353 

0.04328 


1. 1074 

067 
061 

054 
048 


0.04302 
0.04277 
0.04252 
0.04226 
0.04201 


I.I04I 

035 
028 
022 
016 


0.04175 
0.04150 
0.04125 
0.04099 
0.04074 


1. 1009 

003 

1.0996 

990 
983 


0.04048 
0.04023 
0.03998 
0.03972 
0-03947 


1.0977 
971 
964 
958 
951 


0.03922 
0.03896 
0.03871 
0.03845 
0.03820 


1.0945 

939 
932 
926 
919 


0.03795 
0.03769 

0.03744 
0.03719 
0.03693 


1.0913 
907 
900 

894 


0.03668 
0.03643 
0.03617 
0.03592 
0.03567 


3881 

875 
869 
862 


0.03541 
0.03516 
0.03490 
0.03465 
0.03440 


1.0850 
843 
837 
831 
824 


0.03414 
0.03389 
0.03364 
0.03338 
0.03313 


1.0818 
812 
805 
799 
793 


0.03288 
0.03262 
0.03237 
0.03212 
0.03186 


1.0786 
780 

774 
768 
761 


0.03161 
0.03136 
0.031 10 
0.0308^ 
0.03060 
0.03034 


I.07S5 
749 
742 
736 
730 
724 


Nat.  Cos  Log.  d.  Nat.  Sin  Log.    d.  Nat.  Cot  Log.  c.d.  Log.TanNat.    / 

470 


43 

D 

f 

Nat.  Sin  Log.  d. 

Nat.  Cos  Log.  d. 

Nat.TanLog. 

c.d. 

Log.  Cot  Nat. 

0 

68200  9.83378 

14 
13 

73135  9.86413 

93252  9.96966 

25 

0.03034  1.0724 

60 

I 

221  9-83392 

116  9.86401 

306  9.96991 

0.03009   717 

S9 

2 

242  9-83405 

096  9.86389 

360  9.97016 

0.02984   711 

58 

3 

264  9.83419 

076  9.86377 

415  9-97042 

25 
25 

0.02958   705 

57 

4 
5 

285  9-83432 

056  9.86366 

12 

469  9.97067 

0.02933   699 

56 
55 

68306  9-83446 

73036  9.86354 

93524  9-97092 

0.02908  1.0692 

6 

327  9-83459 !  \l 

016  9.86342 

578  9.971 18 

25 
25 
25 
26 

0.02882   686 

54 

7 

349  9-83473  ,  .z 

370  9.83486  :  ;3 

72996  9.86330 

633  9.97143 

0.02857   680 

53 

8 

976  9.86318 

688  9.97168 

0.02832   674 

52 

9 

391  983500 

-t 

957  9-86306 

11 

742  9-97193 

0.02807   668 

51 
50 

10 

68412  9.83513 

13 

72937  9-86295 

93797  9-97219 

0.02781  1.0661 

II 

434  9-83527 

13 
14 
13 

917  9.86283 

852  9.97244 

25 

% 

0.02756   655 

49 

12 

455  9-83540 

897  9.86271 

~. 

906  9.97269 

0.02731   649 

48 

13 

476  9-83554 

877  9.86259 

" 

961  9.97295 

25 
25 

26 

0.02705   643 

47 

14 

497  9-83567 

857  9.86247 

12 

94016  9.97320 

0.02680   637 

46 
45 

15 

68518  9.83581 

14 

72837  9.86235 

94071  9.97345 

0.02655  1.0630 

lb 

539  983594 

^6 

817  9.86223 

125  9-97371 

25 

0.02629   624 

44 

17 

561  9.83608 

14 

797  9-8621 1 

'I 

180  9.97396 

0.02604   618 

4S 

i8 

582  9.83621 

^3 
13 
14 
13 
13 
14 
13 
14 
13 
13 

777   9.86200 

235  9.97421 

25 

oA 

0.02579   6X2 

42 

19 

603  983634 

757  9-86188 

12 

TO 

290  9-97447 

25 

0.02553   606 

41 

20 

68624  9.83648 

72737  9.86176 

94345  9-97472 

0.02528  1.0599 

40 

21 

64s  9.83661 

717  9.86164 

12 

400  9.97497 

0.02503   593 

39 

22 

666  9.83674 

697  9.86152 

455  9-97523 

25 
25 
25 
26 

0.02477   587 

38 

23 

688  9.83688 

677  9.86140 

/" 

510  9.97548 

0.02452   581 

37 

24 

709  9-83701 

657  9.86128 

12 

565  9-97573 

0.02427   575 

36 
35 

25 

68730  9-83715 

72637  9.861 16 

94620  9.97598 

0.02402  1.0569 

2b 

751  9-83728 

617  9.86104 

~ 

676  9.97624 

25 

0.02376   562 

34 

27 

772  9-83741 

597  9-86092 

~ 

731  9-97649 

0.02351   556 

33 

28 

793  9-83755 

^4 
13 

577  9-86080 

786  9.97674 

26 

0.02326   550 

32 

29 

814  9.83768 

557  9-86068 

12 

841  9.97700 

25 

25 
26 

0.02300   544 

31 
30 

30 

68835  9-83781 

13 
14 
13 
13 
13 
14 
13 
13 
13 
14 
13 
13 
13 
14 
13 
13 
13 
13 

72537  9.86056 

94896  9.97725 

0.02275  1.0538 

31 

857  9-83795 

517  9.86044 

952  9-97750 

0.02250   532 

29 

32 

878  9.83808 

497  9-86032 

95007  9-97776 

25 
25 
25 
26 

0.02224   526 

28 

33 

899  9.83821 

477  9.86020 

062  9,97801 

0.02199   519 

27 

34 

920  9.83834 

457  9-86008 

12 
12 

118  9.97826 

0.02174   513 

26 
25 

35 

68941  9.83848 

72437  9-85996 

95173  9-97851 

0.02149  1.0507 

3^ 

962  9.83861 

417  9-85984 

229  9.97877 

25 
25 
26 

0.02123   501 

24 

37 

983  9-83874 

397  9-85972 

" 

284  9-97902 

0.02098   495 

23 

3a 

69004  9.83887 

377  9-85960 

340  9.97927 

0.02073   489 

22 

39 

025  9.83901 

357  9-85948 

12 

395  9-97953 

25 

0.02047   483 

21 

40 

69046  9.83914 

72337  9-85936 

95451  9-97978 

0.02022  1.0477 

41 

067  9.83927 

317  985924 

10 

506  9.98003 

0.01997   470 

19 

42 

088  9.83940 

297  985912 

562  9.98029 

25 
25 
25 
26 

O.01971   464 

18 

43 

109  9-83954 

277  9-85900 

618  9-980M 

0.01946   458 

17 

44 

130  9-83967 

257  9.85888 

12 

673  9.98079 

0.01921   452 

16 

15 

45 

6915 1  9.83980 

72236  9.85876 

95729  9-98104 

0.01896  1.0446 

4b 

172  9.83993 

216  9.85864 

TT 

785  9-98130 

25 

0.01870   440 

14 

47 

193  9.84006 

196  9.85851 

^3 

841  9.98155 

0.01845   434 

13 

48 

214  9.84020 

^4 
13 
13 
13 
13 
13 
13 
14 

176  9-85839 

I^ 

897  9.98180 

0.01820   428 

12 

49 

235  9-84033 

156  9.85827 

12 

952  9.98206 

25 
25 

0.01794   422 

II 
To 

50 

69256  9.84046 

72136  9.85815 

96008  9.98231 

0.01769  1.0416 

51 

277  984059 

116  9.85803 

064  9.98256 
120  9.98281 

0.01744   410 

9 

52 

298  9.84072 

095  9-85791 

O.01719   404 

8 

53 

319  9-84085 

075  9-85779 

176  9.98307 

25 
25 
26 

0.01693   398 

7 

54 

340  9.84098 

055  9.85766 

13 

12 

232  9-98332 

0.01668   392 

6 
"5" 

55 

69361  9.841 12 

72035  9.85754 

96288  9.98357 

0.01643  1.0385 

5b 

382  9.84125 

13 

015  9-85742 

344  9.98383 

25 
25 
25 
26 

0.01617   379 

4 

57 

403  9.84138 

13 
13 
13 
13 

71995  9-85730 

12 

400  9.98408 

0.01592   373 

3 

5a 

424  9.84151 

974  9.85718 

457  9-98433 

0.01567   367 

2 

fo 

445  9-84164 

954  9-85706 

13 

513  9-98458 

0.01542   361 

I 

466  9.84177 

934  9-85693 

0.01516   355 

0 

Nat.  Cos  Log.  d. 

Nat.  Sin  Log.  d. 

Nat.  Cot  Log. 

C.d. 

Log.TanNat. 

r 

46' 


'  Nat.  Sin  Log.  d. 


44° 

Nat.  Cos  Log.   d.  Nat.TanLog.lc.d.  Log.  Cot  Nat 


69466 
487 
508 
529 
549 


9.84177 
9.84190 
9.84203 
9.84216 
9.84229 


69570 

591 
612 

633 

654 


9.84242 

984255 
9.84269 
9.84282 
9-84295 


69675 
696 
717 
737 
758 


9.84308 
9.84321 
9.84334 
9.84347 
9.84360 


69779 
800 
821 
842 
862 


9-84373 
984385 
9.84398 
9.8441 1 
9.84424 


69883 
'904 

925 
946 
966 


9-84437 
9.84450 
9.84463 
9.84476 
9-84489 


69987 

70008 

029 

049 

070 


9.84502 

9-84515 
9.84528 
9.84540 
9.84553 


70091 
112 
132 

153 
174 


9.84566 

9.84579 
9.84592 
9.84605 
9.84618 


70195 
215 
236 

257 
277 


9.84630 
9.84643 
9.84656 
9.84669 
9.84682 


70298 
319 
339 
360 

381 


9.84694 
9.84707 
9.84720 

9-84733 
9-84745 


70401 
422 
443 
463 

484 


9.84758 
9.84771 

9.84784 
9.84796 
9.84809 


70505 

525 
546 
567 
587 


9.84822 

984835 
9.84847 
9.84860 
9-84873 


70608 
628 
649 
670 
690 
711 


9.84885 
9.84898 
9.8491 1 
9.84923 
9.84936 
9.84949 


71934 
914 
894 
873 

853 


9-85693 
9.85681 
9.85669 
9.85657 
9.85645 


71833 
813 
792 
772 
752 


9-85632 
9.85620 
9.85608 
9-85596 
9-85583 


71732 
711 
691 
671 
650 


9-85571 
9-85559 
9-85547 
9-85534 
9.85522 


71630 
610 
590 
569 
549 


9.85510 
9.85497 
9-85485 
9-85473 
9-85460 


71529 
508 
488 
468 
447 


9.85448 
9-85436 
985423 
9.8541 1 

9-85399 


71427 
407 
386 
366 
345 


9-85386 
9-85374 
9.85361 

9-85349 
9-85337 


71325 
305 
284 
264 
243 


9.85324 
9.85312 
9.85299 
9.85287 
9-85274 


71223 
203 
182 
162 
141 


9.85262 
9.85250 
9.85237 
9-85225 
9.85212 


71121 
100 
080 
059 
039 


9.85200 
9.85187 

9-85175 
9.85162 
9.85150 


71019 

70998 

978 

957 

937 


9-85137 
9.85125 
9.851 12 
9.85100 
9.85087 


70916 
896 
875 
855 
834 


9.85074 
9.85062 
9.85049 

9.85037 
9.85024 


70813 
793 
772 
752 
731 
711 


9.85012 

9.84999 
9.84986 
9.84974 
9.84961 
9.84949 


96569 
625 
681 
738 
794 


9.98484 
9.98509 
9-98534 
9-98560 
9-98585 


96850 
907 

963 

97020 

076 


9.98610 

9.98635 
9.98661 
9.98686 
9-9871 1 


97133 
189 
246 
302 

359 


998737 
9.98762 

9-98787 
9.98812 
9-98832 


97416 
472 
529 
586 

643 


9.98863 
9.98888 
9.98913 

9-98939 
9.98964 


97700 
756 
813 
870 

927 


9.98989 
9.99015 
9.99040 
9-99065 
999090 


97984 

98041 

098 

155 
213 


9.991 16 

9-99141 
9.99166 
9.99191 
9.99217 


98270 
327 
384 
441 

499 


9.99242 
9.99267 
9.99293 
9.99318 

9-99343 


98556 
613 
671 
728 
786 


9.99368 

9-99394 
9.99419 

9-99444 
9.99469 


98843 
901 
958 

99016 

073 


9-99495 
9.99520 

9-99545 
9.99570 
9.99596 


99131 
189 
247 

304 
362 


9.99621 
9.99646 
9.99672 
9.99697 
9.99722 


99420 
478 
536 
594 
652 


9.99747 
9.99773 
9.99798 
9.99823 
9.99848 


99710 
768 
826 
884 
942 

lOOOO 


9.99874 
9.99899 
9.99924 

9-99949 
9-99975 
0.00000 


0.01516 
0.01491 
0.01466 
0.01440 
0.01415 


1-0355 
349 
343 
337 
331 


0.01390 
0.01365 
0.01339 
0.01314 
0.01289 


1.0325 
319 
313 
307 
301 


0.01263 
0.01238 
0.01213 
0.01188 
0.01162 


1.0295 
289 
283 

277 
271 


0.01137 
0.01112 
0.01087 
0.01061 
0.01036 


1.0265 
259 
253 
247 
241 


O.OIOII 

0.00985 
0.00960 
o.oo93g 
0.00910 


1.0235 
230 
224 
218 
212 


0.00884 
0.00859 
0.00834 
0.00809 
0.00783 


1.0206 
200 

194 
188 
182 


0.00758 
0.00733 
0.00707 
0.00682 
0.00657 


1.0176 
170 
164 
158 
152 


0.00632 
0.00606 
0.00581 
0.00556 
0.00531 


1.0147 
141 

135 
129 
123 


0.00505 
0.00480 
0.00455 
0.00430 
0.00404 


1.0117 
III 
105 
099 
094 


0.00379 
0.00354 
0.00328 
0.00303 
0.00278 


.0088 
082 
076 
070 
064 


0.00253 
0.00227 
0.00202 
0.00177 
0.00152 


1.0058 
052 

047 
041 

035 


0.00126 

O.OOIOI 

0.00076 
0.00051 
0.00025 

0.00000 


1.0029 

023 
017 
012 
006 

000 


Nat.  Cos  Log.  d. 


Nat.  Sin  Log.  d. 

4^ 


Nat.  Cot  Log. 


c.d.  Log.TanNat. 


■d  ■ 


/ 


/ 


UNIVERSITY  OF  CAUFORNIA  LIBRARY 


